I opened bitcoin-qt after not mining for a couple months and found that I suddenly have no more bitcoins! And I see that there were 2 transactions that I did not authorize sending my coins to other addresses! Is there anything I can do to get my coins back?
Probably not. You can try to look at the addresses the coins were transferred to (or, if those were since sent elsewhere, at those addresses too) and see if you can figure out who they belong to. If the thief is somewhat competent, however, it's a good bet that he's made it very hard to track. If you can determine the identity, you might be able to take them to a court (small claims, civil, criminal...I don't know, and might depend on how many bitcoins were stolen, and where in the world you and your thief live) to demand the bitcoins (or equivalent value) back. But as far as the Bitcoin protocol itself goes: no, you cannot reverse the transaction.
To prevent this in the future, make sure that your computer is free of malware, always encrypt your wallet with a secure password, and consider using an offline wallet. See bitcoin.org's Securing your wallet page for other tips.
Are you sure that the addresses don't belong to you, maybe in some online service you used and forgot about?
No, sorry but "fortunately" you can't do anything, otherwise people could pay for goods and services and then get back the bitcoin used (it's called double spending).
The analogy with fiat money is when someone picks your pocket, asking your bank for a refund for the cash you had won't help.
Some best practices for achieving good security are:
- take care about system password for online wallet
- install official/trusted bitcoin wallet software
- use offline wallets if you hold many bitcoins (create the wallet with an off-line computer!)
- do not put many bitcoin in a single exchange because they control your money (mtgox), or wait for distributed exchange