Un verre de vin/bière : a glass of wine/beer
Perhaps because ‘verre’ ends in e it is easy to imagine it is a feminine noun but it isn’t (‘un’ not ‘une’). Apparently, all over France, waiters sigh when we foreigners ask for ‘une’ verre de vin/bière.
Un autre verre/encore un verre svp*
A cousin once told me that, if one wants another drink, it is better to say ‘encore un verre, s’il vous plaît*’ because ‘un autre verre’ can imply dissatisfaction. However, a French friend recently queried that this is probably better the other way round, so my guiding hand is deserting you here. Just remember it’s best to smile and add ‘merci/thank you’. And never forget to say ‘Bonjour Madame/Monsieur’ before uttering a single word at all.