MCMXCIVRS
More toys than space
Getting back to the original questions, changing the chain and sprockets is a pretty simple task if you possess the tools and basic skills. I also just recently changed the chain and sprockets on my F800GS in about an hour.
There is no need to drop the swingarm as was noted so long as you don't buy the factory chain which come pre sized and riveted. There are lots of good quality aftermarket chains available (don't cheap out) that will fit your bike. They will likely be a few links longer than needed, so you will need to "break" the un-needed links from the chain to get the correct length (just count how many you have now and match it up). You will also need to break the current chain to remove it. The sprockets are pretty straight forward to remove and install, you will need to remove the rear wheel for the rear sprocket.
As for changing gearing, it all depends on what you want to achieve. I have gone down one tooth on the front sprocket of my F800GS (from 16T to 15T) to reduce the gearing for easier off road riding. I don't find any major impact on top end speed (not a speed demon anyway), nor on fuel economy. As noted before, one tooth on the front will not affect the chain length, you can take up the little extra on the adjustment. To affect the same change in gearing by changing the rear sprocket takes a greater size change and would more likely affect the chain length required. You could change both sprockets to find a mid point ratio change if you wanted, keeping in mind that you might have to mess with the chain length by a link or two.
For the tools, get yourself a good quality chain tool set making sure it will perform all the required tasks of breaking the chain (which is pressing out the pins to separate the links), pressing the side plate on, and then finally riveting the pins to hold the plate in place. There are different types of pins which require different riveting styles, so get a tool that can do both (hollow nosed or soft nosed pins) or at least get the tool for whichever type you end up getting. Also good to be sure the tool is for the correct size of chain which is 5xx for your bike. While this is the most common size chain and thus, chain tool, there are other sizes around in smaller bikes and the tools for those are sized accordingly. If you want a tool recommendation, I have the Motion Pro PBR set and it works very well, is good quality and reasonably priced.
There is no need to drop the swingarm as was noted so long as you don't buy the factory chain which come pre sized and riveted. There are lots of good quality aftermarket chains available (don't cheap out) that will fit your bike. They will likely be a few links longer than needed, so you will need to "break" the un-needed links from the chain to get the correct length (just count how many you have now and match it up). You will also need to break the current chain to remove it. The sprockets are pretty straight forward to remove and install, you will need to remove the rear wheel for the rear sprocket.
As for changing gearing, it all depends on what you want to achieve. I have gone down one tooth on the front sprocket of my F800GS (from 16T to 15T) to reduce the gearing for easier off road riding. I don't find any major impact on top end speed (not a speed demon anyway), nor on fuel economy. As noted before, one tooth on the front will not affect the chain length, you can take up the little extra on the adjustment. To affect the same change in gearing by changing the rear sprocket takes a greater size change and would more likely affect the chain length required. You could change both sprockets to find a mid point ratio change if you wanted, keeping in mind that you might have to mess with the chain length by a link or two.
For the tools, get yourself a good quality chain tool set making sure it will perform all the required tasks of breaking the chain (which is pressing out the pins to separate the links), pressing the side plate on, and then finally riveting the pins to hold the plate in place. There are different types of pins which require different riveting styles, so get a tool that can do both (hollow nosed or soft nosed pins) or at least get the tool for whichever type you end up getting. Also good to be sure the tool is for the correct size of chain which is 5xx for your bike. While this is the most common size chain and thus, chain tool, there are other sizes around in smaller bikes and the tools for those are sized accordingly. If you want a tool recommendation, I have the Motion Pro PBR set and it works very well, is good quality and reasonably priced.