My Turkish guidebooks suggest that for a fun experience, I visit a tea house and play backgammon (tavla) with local people. What is the proper way for me (a male) to challenge someone to a game of tavla? And, is it customary to make a small wager, maybe çay veya kahve ?

Istanbul may not be the best place for this. If you are going to the country, you could, at a kahve or kahvehane, when you see an open or maybe closed backgammon set on a table with someone is sitting at that table, go to him and propose, "tavla oynayabilir miyiz?". He may propose a wager of a cup of tea or coffee. They will usually play to three or five, with the first reaching three or five winning. There is no doubling unless you are into serious gambling. Most coffeehouse players will be able to roll at least one die to come the number they want and it will be impossible to get them to use a cup or something for rolling. So, you should practice that also.

I was 17 when i could get what i wanted on one die and feeling cocky about it, when i met an old man (45 possibly
at a coffeehouse in the small town of Edremit where i had hitchhiked in search of an American exchange student whom i had met in istanbul and who was staying with a Turkish family in Amasya which had gone on vacation and was refusing to allow me to visit.

Having arrived late in the evening and found this terrible but cheap bed in a room with six beds, i had sauntered to the kahve near the hotel entrance and watched a game. One of the players left and the other player asked me if i wished to play. I stupidly said yes i play but i am too good with the dice and it may not be fair to him.

Well, he could roll any combination he wished. Killed me and told me to take his advice and not shoot my mouth before i have observed the opponent, in fact to never brag about anything.

Anyway, good luck.