If you've not got a lot of luggage, then BART is definitely the way to go. The AirBART shuttle bus stops are well-marked (one in front of either terminal).
Put two bucks in the fare collector (just like a regular bus), and enjoy that 10 minute ride to the Coliseum/Airport Station. When you enter the station, to your immediate left are a row of BART ticket machines, which take currency or credit cards (however, if using a credit card, be aware that you will be charged for a $20 ticket). Don't let your luggage out of your sight...it's a dicey area at times, and thieves are on the lookout for newbies concentrating on the BART ticket machines.
You should go ahead and spring for a single ticket with enough fare for a round-trip back to OAK
, that way you don't have to worry about getting a ticket when coming back. Any train will take you to SF
EXCEPT the Richmond-bound trains, and the platforms are well-marked.
Get off at Powell Street Station. There should be signs denoting which station exit goes to Union Square (hint: Stockton Street is a closer walk than Powell Street, of course depending on where in Union Square your hotel is located).