So I'm looking to get some bitcoins securely, but I'm not sure how to go about it.
Whatever site I use, how can I be sure that they won't run off with my bitcoins like Mt. Gox?
If they control the private keys, you cant.
I just read recently that there may be issues with blockchain.info. I'd like to know how to get bitcoins without having to trust ANYONE.
I like the way you think, you will fit right in. cooldgamer already gave you one example (electrum), but there are others[1]. They have their pro's and con's as most things. If you have the time and like to experiment a bit you should try most of them and see which wallet fits your needs best. That way you dont even have to trust anyone of us with a suggestion which wallet is good. At the very least you should read the little pop up boxes for each of them. Full validation (bitcoin core & armory) requires you to download and verify the entire blockchain (>40 GB) which might take a while. Keep that in mind.
This is one of the most appealing aspects of bitcoin, but I haven't been able to figure out how to do it. Everywhere I look, it seems like I'd just be opening a "bank" account that would hold bitcoins for me. Maybe I could then transfer them to a different "bank", but how do I get the "cash". (I think this would mean having the address and private key, but I don't know if this is totally correct.)
Yes, if you are the only person that is in control of the private key(s) you can consider yourself holding the "cash". Most wallets hide the private keys from you e.g. in a encrypted with a password inside a wallet file.
(This seems like a simple question that must have been covered before, but I haven't found anything by searching.)
Thanks for your help.
[1]
https://bitcoin.org/en/choose-your-wallet