Can I reinstall or upgrade Android? At the moment I have a rooted Samsung GT-S5570. After I rooted it, I noticed that it started to be slower than it used to be.

I have had this phone for over two years. I update my computer's operating system once a year. I am thinking of doing something similar with my smartphone.

At the moment I am running Android 2.2.1. I think that an upgrade would not be ideal due to old hardware. What is the best option? To reinstall Android 2.2.1 or upgrade to a newer version?

In either case, how do I do it?
If you just want to "refresh" it, you could try a factory-reset, which will essentially revert it to a "blank state" (i.e. remove all user-data and user installed apps), so you can start over with a fresh system. You should however create a good backup beforehand – which is not an easy task with Android prior to ICS (pre-4.0). On the other had, this can be done without any "modifications" (such as rooting or the like).

If you consider updating your device, take a closer look at custom roms like CyanogenMod. There might be other ROMs available which are tuned specifically for your device, and thus giving better performance even with a newer version of Android. This installation process however is a bit more "technical", and in most cases requires rooting the device and installing a custom recovery.