I'm trying to find any clear information on the public transport options from the Galata Bridge in Istanbul to the Esenler Otogar. What options exist, if any?
By far the easiest option would be to catch the Tram and then transfer to the Metro.
From the Galata bridge there are two options for the tram depending on which side of the river you are on - either Eminonu (on the south side) or Karakoy (on the north side), however both are on the same tram line. Both stops are easy to find as the tram crosses the bridge so you can just follow the tram lines. From there, you want to catch the "T1" Tram headed to Bagcilar.
There are two options for transferring to the Metro. The best option is to catch the tram as far as "Zeytinburnu", and then transfer to the M1 metro in the direction of Aksaray. The transfer at Zeytinburnu is very easy as the tram and the metro are together.
The other option to catch the tram to "Yusufpasa", and then transfer to the "M1" Metro headed towards either Magcilar Meydan (M1b) or Ataturk Havalimani (M1a). However this involves a ~500 metre walk from the tram stop to the underground metro station so although it's potentially quicker overall I wouldn't normally recommend it.
Finally you want to disembark the metro at the "Otogar" station. There is a second station called Esenler, but presuming you want the actual bus station this is not the stop you want.
Note that both the tram and metro can be relatively busy, so especially if you have luggage you should plan for it to take a little extra time. Depending on how you pay and which route you take the entire journey will cost between 2 and 6 YTL.
Tram and Metro, as mentioned in the other answer, is one convenient option. But you can also take the bus. IEET bus number 910 is going from Eminönü to the Otogar. A single trip costs 4 YTL or 1.95 YTL if you have an "Istanbulkart" or an "Akbil".
The Eminönü bus station is just next to the Galata bridge.
Advantage: You don't have to change and it is cheaper because you only need one ticket
Disadvantage: Less frequent than tram and metro