First of all, I assume your speed problems are always present? Or can you pinpoint certain circumstances, like being in the range of many overlapping WiFi hotspots, being in the shower, or he moon being in the third house of the capricorn? (Only semi-kidding here: try to think out of the box, but don’t open your mind too much, or your brain will fall out.)
Let’s check the easy parts first: try hard rebooting and check if the problem persists. Probably try rebooting multiple times. It might even be interesting to disable WiFi and take out your SIM, but that might also be tech-superstition.
If the problem persists, try clearing the cache (e.g., in factory mode or with the help of TitaniumBackup) and see if the problem persists. If so, try to find out in the apps menu which apps are running, and if they take up much memory. Try clearing the cache of some apps which are definetly lagging, e.g. the Camera app, the e-mail app, Whatsapp, etc. If the problem persists, try to remember which apps you installed last, and check if de-installing is an option to try out if the problem goes away.
If this does not yet lead you to salvation, please provide us with some more info: the OS version you are running, if you have encrypted your phone or not, if you have installed Google Apps, if you have applied the unified storage update, if you are running your apps from a SD card or if they are stored in the internal storage (or, if you did not apply the unification, in the phone storage). It would also be interesting if you phone’s memory is especially full, or you have transfer hundereds of MB to it on a regular basis (e.g., lots of music or films every now and then.)
We still might find some software problem. Even the screen flickering might be caused by something in software. Something of a last resort would be a factory reset. And if that does not help, you can still send your phone to FP support and let them find out if it’s a hardware-related problem. They are really customer friendly (I got my phone fixed on warranty twice by now).
Disclaimer: Whatever you do to your phone, it’s worth running a backup first. And don’t use the preinstalled Backup and Restore app for a proper backup.