Bangkok is a huge city. So forget to think you will be able to properly orientate yourself within a day or two.
But there is a trick. Most places that are important for you are nearby a Sky train (BTS) of Metro (MRT) station.
Most important for you will be the Sky train since this is the oldest mass transport system and connects most places of touristic interest; the MRT (subway) was added later and connects more non touristic places as well. Due to this I recommend you to get used to the Sky train BTS Map and use this as an orientation method in Bangkok.
Easy Orientation Skytrain map BTS Skytrain on elevated rails
You can either buy a ticket every time you do a travel or you buy a smartcard which is a prepay card that makes traveling much more convenient. You then place the smart card at the entrance and exit gate and the amount is automatically deducted. A single Ride costs between 15 and 45 Baht. A One day BTS smart pass costs you 120 Baht and an unlimited valid smart pass (prepaid Card, every ride is deducted from your purchased amount) costs you 100 Baht plus any amount of Cash you want to charge the card with.
Unfortunately you will need different tickets for the Airport Link, BTS (Skytrain) and MRT (Metro). Most tourists will use the Airport Link plus the Skytrain, so buy a single Airport Link ticket plus a BTS Skytrain Pass.
There are two BTS lines operating, the Silom Line and the longer Sukhumvit Line. Only interchange is the SIAM Interchange station.