It voids your warranty if you open up the console. I've tried fixing an Xbox 360 myself by doing the X-Clamp replacement but that fix only lasted about 8 months before it red ringed again. Since Microsoft paid 1.5 billion dollars for you to have a 3 year warranty for your Xbox 360, you should just have them fix it for you for free. That is, assuming you've had this Xbox 360 less than 3 years.
Considering the age of this 360, and the cost of out-of-warranty repairs, it's likely not worth it for you to get this particular 360 fixed via Microsoft - for that cost, you might as well just buy a new one.
There is, however, another option - fix it yourself. It may sound intimidating, but the fix isn't particularly complex, and requires maybe $15-$20 USD in parts. There are a few good guides if you Google a bit - here and here aren't bad, but there's a real famous PDF floating around somewhere (I'll take a look around and see if I can't find the name of it for you).
I've performed the heatsink-clamp replacement fix on four different RROD'd 360s, all successful. Especially due to the age, you're not likely looking at an E74 error, so you've got a fairly good chance this will work for you (and worst cast situation, you're down a little bit of money).
Whatever you do, please keep in mind that you can sell these for parts online - there are people out there who will buy these and fix them up, so if you're going to get rid of it, at least make some money to recoup the cost of buying another 360 new.
EDIT: As @Dave mentioned, don't forget to check the 360 support site. I've had numerous friends call me about RRODs, only to tell them to check all the connections to their power supply and have it start working again.
Best Answer
So the first step is to understand what the Red Ring of Death actually is caused by. This particular error is very broad, and can be caused by anything failing on the motherboard, such as bad soldering, bad heatsink application, to poor parts. So unfortunately, a lot of times it's not a matter of if you'll red ring, but when. Now there are some options which I'll detail in order of cheapest to most expensive.
Cheap Solution
Nice and free however not really reliable. Make sure your Xbox has lots of room to breathe, keep it in a cool place with lots of airflow, and don't leave it running for long periods of time. This will minimize your chances of something going wrong. However a red ring can still occur, but this will give you the best chance of preventing it.
Moderately Priced Solution
Here's where things can get a little tricky. If your warranty has expired (or if you really don't want to deal with Microsoft's warranty system, then pop that sucker open and add some aftermarket stuff, (Rims, spoiler, some rad decals!). Seriously though, there are some aftermarket solutions that will help your Xbox out, and won't cost an arm and a leg. I'll go over two types of products, but there are always more out there.
Expensive Solution
Expensive yes, but also quite simple. Buy a new Xbox. Yeah, it's pretty obvious but it does work. The newer Xboxes (I can't attest to the Xbox slim, but the newer old Xboxes this applies to) have better parts, and are less likely to fail. Plus having that manufacturers warranty is pretty nice too.
So there you have it, options are always good.