I'd like to note that paging has nothing to do with how much hard drive space you have available (at least not directly; see below), but rather has to do with how much RAM you have. Very different. When you run out of RAM, the system has to store excess RAM somewhere to prevent the whole system from crashing, and typically the only choice is the hard drive, which is slow. Hence, paging (or swapping, as it's called in Linux). If this is happening, you're pretty much certain to be running slow no matter what, so one possible solution is to upgrade your RAM. Alternatively, you can try to reduce your RAM usage.
Hard drive space can technically affect speed, but the mechanism is unrelated to RAM. Rather, what can theoretically happen is as you lose hard drive space, your data can become more fragmented on the physical hard drive, which can make access times longer. It's like if you have a full notebook, so you add new notes to scattered blank spaces in pages throughout; doing so means you have to look at more pages and flip through more to find the right ones, which makes you take longer to read the notes.
That being said, I said "theoretically" because this is unlikely in a modern system. Modern filesystems like ext4 and NTFS are designed to minimize fragmentation in the background.
"Too many programs installed" shouldn't affect speed, but too many programs running at one time certainly can. This is more or less because programs have to take turns using shared resources, such as the CPU and the hard drive.
Ultimately, though, there's no easy answer for why a computer might be slow. Fixing a slow system can involve all sorts of interventions: using different software, installing new hardware upgrades, replacing defective components, etc. It all depends on what's going on and what your needs are.
But yeah, check your RAM usage and see if all of your RAM is being used up. If so, that's probably the culprit.
Another thing you might want to check is your cooling system. If the fan on your computer is too dusty, that can make it less effective at keeping the CPU cool, which will force it to work less hard. You can fix that very easily with a little compressed air.