"Getting slow" can be caused by a couple of different things -- separately or combined. It could be an app hogging your CPU, or the system running low on free RAM. It's even possible that some service runs havoc, blocking resources (though this one would be a bit harder to track).
Easy tracking for beginners (and for free)
There are several possibilities to find out what's going on. The easiest is to install e.g. OS Monitor and, when the device gets slow, starting this app and taking a look at the running processes. First order them by CPU usage: Is there any app constantly using 50% or more CPU? If so, you probably found the evil-doer. Next order them by RAM usage: Any app "eating" extra-ordinarily much?
Don't worry to much about how much RAM is free. There's no such thing as "unused RAM being a good thing" -- neither on Android nor on Linux. Free RAM will be used to speed things up, e.g. by caching files. Of course, if more RAM is used, those caches get freed.
Monitoring the system
If the easy approach does not help, you need heavier armament -- and a little background knowledge to use such. There's e.g. SystemPanel which can monitor your system. More useful is the paid version, which can do so in the background, and also saves its monitored data -- so you could even check after a reboot. An alternative helper-app is System Tuner (here again the paid app is more useful). Also, Tracing Memory might be worth a look (can't hurt, it's free).
When you found out the bad app, actions depend on what it is. Is it an app that came pre-installed, or even belongs to the Android system, you e.g. could not simply remove it. Check the app's configuration, contact the developer, or search the web for known trouble with the evil-doer are always good steps to take. Of course, once you found the culprit, you can always come back here and ask a more specific question -- and with the background information you collected, we probably can give you better hints to solve the issue.