Several friends have had computers that declined in performance with age, and assumed they would need to buy a new one. Not all computer performance problems can be fixed by a patient amateur, but many common ones can.
Problems #1 and #2 are laptop specific, but #3 and #4 apply to any computer.
Many of these fixes involve taking the computer apart. Ifixit.com has a lot of good manuals and guides, especially for Apple products, and you can look up the manual for the model of computer you have, which may include instructions for some simple repairs. You can also pay someone to do some of these things for you, though then you might not be saving much money relative to just buying a new computer.
1) A laptop that runs hot and makes noises, and slows down once it heats up.
Sometimes this just means the laptop's internal fans are clogged up with dust. If you can open up the laptop and find the fan units, then get a canister of compressed air and blow away the dust.
Or you can do what I did to my girlfriend's MacBook fans, and take them apart and actually wash them. (If you do this, dry them off before you put them back!) I also oiled the fans with some WD-40.
2) A laptop battery that runs out real fast
Sometimes this means you need to replace the battery - which is itself an easy repair, you can usually order the battery separately - but sometimes it's even simpler than that, it's just a weird software issue. Before you even order a new battery, take the old one out, and then hold down the power button for 30 seconds while the battery is out. Then put the battery back in.
3) When you open certain programs, or too many programs, or browse the web, your computer starts to run slow and take a long time to do things.
There are three major reasons this could be happening. Either your programs are asking the computer to calculate too many things at a time (i.e. your processor is too slow) or they are asking it to keep too many things in short-term memory (i.e. you don't have enough RAM), or you are asking it to read and write too much to its long term memory (a lot of hard disk usage). So the first step is to figure out which one is the problem, since fixing the wrong problem is a waste of both time and money.
Most computers both have utilities that tell you how much of your computer's capacity you're using. I'm going to call this the usage utility in the next paragraph. If you're running Windows this is called "Task Manager". Press Control+Alt+Delete and click on "Task Manager. If you're on a Mac, it's called Activity Monitor, you can get to it through the little Spotlight tool on the top right of your screen. If you're on another OS I don't know what it's called but it almost certainly exists. Whatever machine you're on,
There will be multiple tabs in the usage utility. You should be able to see how much RAM is being used, and how intensively your processor is being used, and how fast your computer is reading from and writing to the hard disk. When your computer is running slow, open up the utility and see whether your RAM is near capacity, or your processor is near capacity, or whether your hard disk is being used a lot.
If your processor is running at 100% capacity, you're out of luck. (If you have multiple processors, you should only be worried if they're ALL running at capacity - otherwise the processors are not what's slowing you down.) You may as well buy a new computer, since replacing a processor is not for amateurs, and if you are reading this, you are probably an amateur.
If your RAM is at capacity, then one of two things could be going on.
a) One program could be a memory hog. Look at your list of active programs in the usage utility. If one is taking up a tremendously disproportionate amount of RAM, and it is not a web browser, then decide whether it is needed. In my case, I noticed that the HP printer utility was taking up all the spare RAM on my computer, and the solution was to force-quit that program whenever I noticed my computer freezing up.
b) You could just need more RAM. Look up what kind of RAM your computer will accept - you can do a web search for this or it should say in the manual (which you can also do a web search for) - and buy the memory cards with the most memory per card that your computer can handle. (Some sellers of RAM have tools that can pick out the optimal combo for you on their websites.) Installing RAM is really easy as far as computer enhancements go - you open up your computer, pull out the old memory card, and stick in the new one.
If your problem is just reading from and writing to the hard disk a lot, then you may want to upgrade to a 7200RPM hard drive, or a Solid State Drive. First check to see if you already have one of these - there should be a way to see some "about my computer" stats. If you don't then you need to figure out whether you're comfortable doing this upgrade.
Depending on your computer, upgrading the hard drive could be easy or it could be hard. The laptop upgrade I did was easy, but I'm not even going to try upgrading my iMac's hard drive - it's just too much trouble to get at. See if there's an online manual or guide that tells you how to replace your computer's hard drive.
If you don't already have a fast hard drive, and you think you can do the upgrade, you need to know which kind you want - a fast hard disk, or an SSD. Basically, the SSD will usually be faster, but you don't get as much storage space. That's never been a problem for me, and you can always get an external hard drive for storage, but if you're using hundreds of gigabytes then that might be an issue for you.
4) A specific component seems like it's not working
This might seem too obvious to list, but I figured I'd say it anyway. If it seems like some specific hardware-related computer function is failing - for example, your optical drive (e.g. CD-RW or DVD reader or Blu-Ray drive) stops working, it might not be a hardware problem. Look up whether there are known software issues with your computer, but if there aren't, then it's probably just a hardware problem, and the repair might be easy. You can often order the exact same component you're replacing.