My computer is currently running with an AMD 1.8ghz X2 processor and 1GB of RAM. I'm thinking the RAM, but which should I upgrade first, the RAM or the processor?
Really, the answer is "What is your current bottleneck?" If you're playing certain video games (and that's what you really want to speed up) the best upgrade might be a new video card. If you do lots of database stuff, the RAM might be a better choice. If you're doing enormous amounts of pure math, say 3D rendering ... you'd have to see what's being overused, I guess, because that's both RAM and CPU intensive. In any case, the answer depends on what you need, which we don't know from the way you phrased your question.
To answer your question, RAM is probably the first thing to upgrade. Remember, you can probably overclock your CPU to 2.0GHz or so. I have a friend who has a much older AMD system than yours and it's overclocked - he says it makes a difference. So, buy RAM, you can't just make more of that RAM space yourself. (OK, swap space, but that is a different story...)
It really depends - (Assuming Windows) Open up task manager, If your Memory usage is high, increasing the memory could give you the "umph" that you need. If however, your memory is in the 45% or lower it is unlikely that adding any more would be noticeable to speed.
I am not to familiar with AMD CPU names, but if it is more than 3-4 years old, you can probably find the highest one that is compatible with your motherboard cheap on eBay which may be worth considering (remember, you will also need thermal paste and/or (as usually included) a new Heatsink and fan).
If you want to get a much newer CPU, it probably would not even fit in the socket or be compatible requiring new memory anyway which will mean you will need to do both.
Answer the most basic question: What motherboard are you running?
No point in worrying about RAM if your board can't support more. Second, what kind of CPU's can your motherboard support? Again, no point in worrying about it if your CPU options are maxed out.
Then, what OS are you running? Certain OS (XP) may benefit from more RAM whereas others like Linux won't really gain that much. The RAM type if old may be expensive also.
Next, what kind of CPU socket types can your motherboard support? If your board doesn't support more advanced CPU's, then carefully consider what you gain by upgrading at all. If you can only go from 1.8 to 2.2, then the increase isn't that significant.
If your computer is rather old, CPU upgrade might be possible, but there's the chance that other components in your computer will not be able to keep up with the performance of the new CPU.
You can get a "cheap" Dell for $400. If you look to upgrade, you might find yourself needing a new motherboard, a new CPU, new RAM, a new GPU and possibly even a new PSU to power everything.
I rather suggest investing in a new computer.