I've got a Boardman Road Sport which I use for commuting and doing reasonable length rides during the week etc. I have Shimano Claris as my rear derailleur and so is some of my groupset. I would like to upgrade my cassette to an either 9 or 10 speed to give me a greater variety of gears. To do this do I need to change my rear derailleur from 8 speed to Shimano 105 or something similar? Or can I just change my cassette? Do I also need to change my shifters as well?
If you want to change your cassette to something with more gears, then you will need to replace the rear shifter and possibly the rear derailleur. A cheaper option would be to get a wider range cassette which still has 8 gears. You'll have bigger jumps between the gears, but you'll have a bigger range of gears. With a wider range cassette, you may need a new rear derailleur with a longer cage.
You can upgrade to as much as 1 speed up from what you have, as long as the rear derailler supports it. To see what your derailler supports, you could have a look at velobase and search after your derailler model. For extra speeds, usually, the most important part is the rear derailler. It does all of the heavy lifting when changing gears. That means that if your derailler's max stretch is not able to encompass the extra cog, you won't be able to shift to that gear and you'll have a dead cog. Also, the length of the shifting cable will possibly need to be changed too.