Never been to the Preakness, but I have been to a handful of others. First, check the weather. Wear the correct gear.
Check out the building. I'd assume that it has a bunch of cool historical stuff. Be ready for lines. Go to the bathroom when it is convenient, because there definitely could be a time where you're doing a dance and it is out the door.
As for betting, I go to OTB (off track betting) at least once a week. I was a math major. I understand odds. My FIL probably nets about $4k a year on this, and has taught me a lot. I do not understand handicapping enough to make a go of it. I do not bet often, and I do not bet much. I'll put in some more comments later in teh year as I can see how individual horses are running and in what races, but you'll basically want to just bet $2 bets for Win, Place, and Show on some of the favorites. Bet a cool name here or there, because it is fun, but do not get carried away. I typically will make about 5-10 bets, and rarely will I spend more than $10 on those bets. I bet the minimum bet allowed on most bets.
The machines are going to be a lot faster than going to the tellers; if you do go to the tellers, tell them Track, Race #, Bet, Number. So if Secretariat is number 3, running at Pimlico in the 4th race, you'd say "Pimlico race 4; $2 on #3 to win". Don't use the names of the horses, it slows everything down.
Common bets:
Win - bet on the horse coming in first
Place - bet on the horse coming in first or second
Show - bet on the horse coming in first, second, or third
Exacta - Bet on the first and second place horses (in order)
Trifecta - First, second, and third (in order)
Superfecta - First, second, third, and fourth (in order)
Box - any order on any of the "acta's". - so say you know that the top 3 horses in a race are going to be 2, 4, and 5, but you don't know what order, you would do a Trifecta Box with 2,4,5.