I had about ten games on Steam on my old computer. Most of them were played at least once. Now, I want them on my new computer. However, I don't care about transferring them, I just want to know what they were. Is there a function in Steam that lets me track my download history or something similar?
Yes, if you have access to your old computer and you haven't yet uninstalled your games/steam yet. If that's gone, I don't belive steam stores any sort of list of everything you've ever downloaded.
On any PC with Steam installed you have access to a list of currently downloaded titles (it's one of the filter options on the Library page, I forget the exact menu item and I don't have Steam here). The only thing I'm aware of that you can do is manually go down this list and download the same games on the new PC (or alternately copy over the whole Steamapps folder and keep your game content and possibly saves).
By simply installing Steam and going to the Library, there should be a list with all the games you own.
If there isn't, still on the Library page there's a search function in the upper left part of the screen. From there, you can access a drop down menu where you can choose what to list.
If you choose "All Games", you should see all the games you own on Steam and can redownload and reinstall whatever you want.