I have a strong GPU, CPU, and enough RAM to choke a goat. I have installed BitCoin Core 64, and my wallet is (finally) synced. What do I do to start mining. Do I just leave the program running or do I have to join a pool like Slush's Pool?
You don't stand a chance. No matter how strong your computer is, you will not make more than a few cents (USD) a day.
There are a few scrypt-based altcoins (e.g. Litecoin) that may still be barely profitable.
Litecoin and other scrypt based coins are fair game with a GPU until the scrypt ASICs start coming out. It's hard to tell exactly how long it will be until one comes out but it shouldn't discourage you from starting to mine, given the delivery time it took BFL and other SHA-256 ASICs to deliver.
Even once that happens, you still have multi-algorithm coins such as quark which can be GPU mined with this software: https://bitcointalk.org/index.php?topic=475795.0
Have a look at Yacoin and Ybcoin too.
And with all this, there are still some coins which are not easy to GPU mine at all, such as Primecoin.
Of course, there is a progression, as a coin gets more popular people will start to develop software and hardware to mine it quicker. But if you mine at the right time you can get a sizable amount of coins before ASICs push up the difficulty (and price).