This is a good question, but it looks to me that the present answers disavow the natural complexity of the terrain that had to be navigated as well as Karl XII's (generally anglicised as "Charles" or "Carolus") complex position in the beginning of his Poltava campaign.
I've decided to structure this as a "concise" summary as well as a longer narrative of the Swedish moves. Mostly based on Peter Englund's 'Poltava' (not specifically quoted) and Angus Konstam's 'Poltava 1709'.
- By 1707, Karl XII of Sweden had defeated his enemies in Denmark, Saxony/Commonwealth, and was ready to face off against Peter I of Russia. While fighting in Saxony, it is suggested that Karl's advisor, Carl Piper, was bribed by the British to invade Russia in opposition to the French who wanted Karl to open another front against the Austrian Emperor. The Swedish King, in any case, preferred to the fight against Russia; at this stage, the Swedish did not reach out to the Ottomans with diplomatic overtures. Karl's army was in Poland at this time, with Swedish Livonia defended by General Lewenhaupt.
- Karl wanted to advance against Moscow from the west through the "river gate" between the Daugava and the Dniepr, passing by Smolensk.
- Russian Prince Menshikov had been sent into Poland to carry out a fighting retreat, destroying any supplies that the Swedish could use for themselves.
- Karl advanced on Smolensk through Lithuania, but found that he could not supply himself as much as needed. Lewenhaupt's supply column from Livonia having not arrived yet, Karl decided to move south and to forage between the Sozh and Desna rivers in Northern Ukraine.
- While moving south, the king learned that Lewenhaupt's column had been defeated and all the much-needed supplies captured.
- Karl decided to move further south into the yet-untouched region of Severia, but failed to capture the territory fast enough and the Russians occupied the major towns.
- Hetman Mazepa declared independence and support for Karl, opening up the chance of provisions in the Ukraine along with a new avenue of approach against Moscow from the south.
- Mazepa was defeated before the Swedish reinforced him and all of his supplies of food and ammunition destroyed.
- The winter of 1708/9 now saw the Swedish King (in the Ukraine, relatively close to Poltava) start with a diplomatic approach towards the Ottomans and the remaining Cossacks in order to garner their support. These moves were countered by the Russians.
- In the spring of 1709, the Swedish and Russians both concentrated forces by Poltava.
- Karl was wounded in June, and unable to direct the army while the Russians crossed the Vorskla and advanced against the Swedish.
Opening Stages of the Great Northern War
When the war started in 1700, the Swedish were fighting a combined alliance of Russia, Denmark, and the Commonwealth/Saxony (as the Duke of Saxony was the elected King of Poland). Karl's opening stroke managed to defeate the Danish and make them sue for peace within five months of the opening of the war. This left Karl Russia and the Commonwealth/Saxony to deal with. Russia was given a nose-bleeding at Narva, 1700, which prompted Peter I to modernise the Russian army while the Swedish decided that the biggest threat at that moment was Saxony.
Karl XII Fighting Saxony
Karl's campaign in Saxony had the aim of defeating the Duke of Saxony, Augustus II, and to depose him from the throne of Poland-Lithuania. After a campaign that stretched for six years and throughout Poland-Lithuania, Karl marched into Saxony, devastated the land, and was able to force Augustus to agree to the Swedish demands in the Treaty of Altranstädt. Karl set up Stanislaus Leszczynski (using my Osprey's name 'format') up as the new King of Poland. This left Russia -- who had been re-training its armies for nigh on five years now -- as the last standing state in the fight against Swedish overlordship in the Northern Baltic.
France was, at this stage, interested in having the Swedes join the War of the Spanish Succession against the Austrian Emperor. Meanwhile, Great Britain with the capable Duke of Marlborough at the head of the allied armies was doing everything possible to ensure the Swedish kept their eyes on Russia. Norrhem's 'Christina och Carl Piper' suggests (based on the WP conclusion) that Carl Piper, a main courtier of Karl XII, was bribed through his wife by Marlborough to support an invasion of Russia and Moscow. Englund does not mention this though his work is also older (1988 to 2010) and more focused that Norrhem's.
Plan to Fight Russia
The Swedish Quartermaster, Gyllenkrok, was one of the main enablers of the following campaign. Gyllenkrok had purchased large numbers of maps of Poland and Russia, but there is no actual original plan of operations that remains because all of the documents were destroyed before surrender at Perovolochna after Poltava had been lost. Nevertheless, Konstam discounts that Karl wanted to move south in his 'Poltava 1709' (Osprey):
Suggestions that either he [Karl] had no plan or that he intended to move south anyway can be discounted.
His main war aims were to free the occupied areas of his Baltic empire, and to achieve a lasting peace that would allow him to return to the main European stage. The former would in preference be achieved without letting the Baltic states become a battleground.
Once on the border of Russia, the options were to head north towards St. Petersburg, forward to Moscow or south into the Ukraine, and then on to Moscow with Turkish help.
Konstam precludes the first and last because Karl's new aims included not fighting in Livonia (preventing devastation in his own lands), minimising fighting in Poland-Lithuania (minimising fighting in now-allied territories), and also because Karl did not at this stage involve diplomatic overtures to the Cosssacks or the Turks.
From nature's point of view, the most preferred route (the same Napoleon took later) would pass between "the river gate" between Daugava and Dniepr. In support of this route is also that the Swedish distributed propaganda leaflets around Smolensk which is the main fortress that defends this passage. The benefit of this was reducing the number of river crossings, with other routes blocked by long north-south rivers.
The Swedish objective, as mentioned above, was to force a peace on Russia to re-focus on Central Europe.
The Russian plan, for their army in Poland, was to withdraw before the main Swedish army. Prince Menshikov was in charge of this force, and he set fire to crops in his path to cause the oncoming Swedes to have problems of supply.
Peter, meanwhile, also had to ensure the protection of St Petersburg which was intended to be his glorious new capital. The Russian main army, therefore, in general maintained a position up to when Karl started advancing which placed it in between St Petersburg and the Swedish force.
The campaign proceeded in small steps with Menshikov's retreat going to plan. Peter joined Menshikov in early 1708, and as the force was insufficient to meet Swedes on the field of battle, the retreat continued through Lithuania. While here, Karl ordered General Lewenhaupt, commander of the Swedish Livonian troops, to meet the main royal army on the Russian border by mid-summer. The Swedish were, however, suffering by not being able to forage as effectively as they needed to.
At this stage, the 1708 campaign was to restart. Count Piper advised the King to advance against Novgorod, but Karl still supported the advance through the "river gate". In short, the Swedish successfully crossed the Berezina stream. They were by the Vabitch on 30th June, and the Russian army on the other side (at Holowczyn) had fortified their position. Karl crossed the stream, defeated the Russian army, and opened the "highway" to Moscow.
The Russians re-fortified themselves further up the main pathway at Gorki while Sweden waited for the Livonian troops to join him. Karl was unable to wait for Lewenhaupt, moved against the Russians who abandoned the field fortifications at Gorki and retreated. Karl tried to capitalise on this and conquer Smolensk, but the Russians moved troops up in defence. The marshy terrain favoured the Russians, and the Swedish decided to move southwards instead of engaging, primarily because they needed more supplies to cross and fight across the scorched earth. Lewenhaupt's column was to bring these items to enable campaigning across the scorched earth landscape.
Desertions from his army were mounting, and there was still no firm news of Lewenhaupt's supply column, which would be needed to cross the scorched landscape. Consequently, the King made the decision to head south, abandoning the advance along the Smolensk-Moscow axis. His main aim now would be to keep his army alive, by beating the Russians to the undamaged region of Severia.
The Swedish had to capture three fortresses in the area (Mglin, Pochep, and Starodub), but only managed one of these with the Russians reinforcing the others. The Swedish army had, by now, lost 13,000 men and had to recover from a long march south.
Meanwhile, Lewenhaupt had been defeated at Lesnaya, losing half his troops and the entire supply train. Lewenhaupt's stragglers joined the royal army at a few weeks after Karl had made it into Mglin (the conquered Severian township).
The Final Path to Poltava
Lesnaya made [a move against Moscow on the Kaluga road] impractical. One other strategy lay open. The hetman of the Ukrainian Cossacks, Mazeppa, had just declared himself for Charles, leading his people in open revolt against the Tsar. ... he had enough support to open another window of opportunity for Charles. ... the King wrote to Mazeppa promising his support and requesting the provision of winter quarters in the Ukraine. With a friendly populace and plentiful supplies, the Ukraine was seen as 'a country flowing with milk and honey'. The plan was now to ... attack Moscow from the south in the spring.
Karl raced for the south, to make it to Mazepa. However, Menshikov made it to the hetman's capital at Baturin before Karl (the Swedes were four miles out), burnt the place, and destroyed the supplies of ammunition and food. Before winter set in, Karl set up his troops in winter quarters while the Russians again went and blocked the roads to the east.
By the start of 1709, Karl's Swedish army was quite close (50km) to Poltava, at Gadyach, though it was strung up in a long line which also made it quite close to Poltava itself. Karl finally used the winter to talk to the Turks and the Cossacks in more diplomatic channels, but the Russians countered these options. In the start of the spring season, Karl decided to concentrate his army while the Russians concentrated their troops on the other side of the Vorskla from Poltava.
Karl was wounded on 17th June, and the army was unable to do much more beyond staying put. Peter re-joined his army in the end of June, and crossed the river while Karl's paralysis meant this was allowed to happen. As the Swedish King had actually been seeking battle in the past, the stage was set for the battle itself.
Researching another topic, I found a tool of the Russian Empire as GIS, and was able to make a more informative graphic than the above. The cities are 1820's so a century later, but the general pathing of the roads should be similar. Piper's recommendation against Novgorod is detailed (not followed) along with the ideal path through the "river gate" as well as where Lewenhaupt should have come from. The later hope of a new attack from the south is also detailed.