Long story short, I'd like to try out a few PC exclusive franchises. Nothing more. My processor is Intel Core i5 3210 CPU 2.50 ghz. Memory is 8gb ram. It's a Dell Inspiron. I know my laptop definitely isn't some top of the line laptop for gaming but for around $600 I'd hope it's at least decent. 

The CPU should be able to run most games in at least medium settings and some games in high. You have plenty of RAM for games if it's nice and fast. However, what's really important is your video card. Most laptops have an integrated video card that shares the system memory. You should still be able to play games with one of those, but if you're lucky you might have a discrete mobile graphics card which will give you more power and it's own RAM.

If you have Intel HD graphics:

The Orange Box
Left 4 Dead/2
Portal 2
StarCraft II (lowest settings)
Diablo III (medium settings)

Most indie games more than likely.
Anything before 2006 you should be able to run fine. As for newer games, turn down those settings ;)

whats the GPU?

Tell us the GPU..

Most likely GT640M, GT 630M, HD 7670M. Those are the common ones.

Doesn't seem like very good specs for newer games honestly... I'm not getting good google results on the CPU. It's very hard to make a laptop do any real gaming well unless you go nearer to a $1000 or way beyond...... some games would probably be OK, especially older, but it would be more important to know the GPU since that'll probably be your bottleneck.

Basically if it's not dedicated, it will rely on your system RAM anyway and if it's the very basic, your frame-rates will be downright painful in some graphics heavy games. FPS's won't like this and you'll probably die more because you miss your shot. At around $600, you generally will only get the basic graphics.. I think I paid $850 to get something beyond basic a few years back. It doesn't make it impossible to game, it just limits what you can reasonably expect out of it....

Staying plugged into the wall and turning off any power saver settings will also help you some. Power saver bottlenecks the CPU to save battery. Also don't block the vents and play on a flat surface so it gets adequate cooling so you're not overheating the machine (don't play on your blanket on your bed for example)... seems like a no brainer, but a lot of people miss this and don't realize you can slow down your computer if it overheats too much or it can even make it shut down.... that tends to happen if your computer runs somewhat hot to begin with. Intel's are generally cooler so it's probably less of a worry for you...

Inspiron is more of a business grade laptop, but the processor is decent. If you have a good GPU (video card), it will be decent at gaming. If you lower some settings, you should be able to have a good experience. For the newer games, like in the last 6 years or so, you'll most likely need to lower the graphics settings, but I think it would suffice, depending on the game. If you wanted to run something like Planetside 2, I'm not sure if your laptop could do it. Maybe on low settings. But what do i know, I haven't even played Planetside 2 yet lol

We need to know the graphics card model.

If it doesn't have a graphics card then, not good.

It's a good processor, but I'm 99% sure that didn't come with a dedicated graphics card. That means you have an intel integrated graphics card. Basically, don't expect to play anything recent, except for indie games.