I'm sorry to say it, but that's not nearly enough money for a computer that will run all TS3 packs on the highest settings. With most games, you could cut corners and merely see a decrease in performance, but with TS3, lower-end hardware may struggle to run more than the base game and a couple of early packs. A $300 computer is even a level below that, where you might not even be able to run the game at all.
Even if you have the best possible luck with sales, you'd still need to spend at least $700 before tax on a laptop, or maybe $600 for a desktop, not including the monitor. I don't know what other games you have in mind, but their requirements may be higher. (You're welcome to list them, if you want to know what kind of hardware they'd need.) Desktops can also be upgraded later, if you want, whereas most laptops won't take more than extra RAM and maybe a second hard drive.
If you do think you can spend a bit more, let me know, and I'll try to find something. Please also list your country, since prices, and the best stores, will vary.