Teknaf is the southern-most city on the mainland in Bangladesh, on the narrow strip running along the coast of Myanmar. It's a dusty featureless frontier town - this is remote Bangladesh and it feels like it. Electricity can be scarce, so having a flashlight for the dark evenings is wise.
Buses running from Cox's Bazar (Tk 70, 3 hours) are crowded and pass through nice scenery as you get closer to Teknaf. If you're trying to head straight to Saint Martins Island and don't want to stay the night in Teknaf, try for a 6AM bus from Cox's Bazar and specify which ferry port you want to get off at - you may make it.
The main ferry port is about 5km north of town on the road to Cox's Bazar. Eagle I and Eagle II leave from here at around 10AM daily.
Keari Sindbad has its own port about 3km further north from the main port on the same road. Their service is a little more reliable and costs about the same. The ferry leaves at 9:30AM daily, then leaves Saint Martins Island around 3:30PM. Roundtrip tickets to Saint Martins Island run Tk 440-540 depending on the part of the boat you prefer.
Auto-rickshaws (babytaxis) can be found near the bus stop. To the main ferry port should costs Tk 50 and to Keari Sindbad's port Tk 70. Cycle-rickshaws may be cheaper but the road north can be steep and you'll end up walking a good deal of the way. Using the Bangla word "Bandar" (port) will yield the best results, but beware that "bandor" means monkey, so choose your vowels carefully.
There's a market down the road behind the bus stop and over the bridge in which can be found goods smuggled from Myanmar, possibly even beer.
There's a largish restaurant tucked behind the bus stop that serves decent food at decent prices, and several more similar restaurants along the road behind the bus stand en route to the bridge. None are signed in English.
Inquire at your hotel or at one of the local restaurants and you may be able to score some beer from Myanmar.
It's likely that you'll have to spend a night here en route to Saint Martins Island, since the ferries leave so early.