Let's face this one simple fact -- we have all at some point of our lives been faced with the existential dilemma of how to address someone in Bangla: 'tumi', 'tui' or 'apni'? Now this is one question that has no particular answer. See, tumi is used when talking to someone close. Tui, on the other hand, can be used either when speaking to somebody REALLY close, or a younger relative, or someone who decidedly belongs to a lower social class (yup, we're like that). Apni shows respect, while simultaneously drawing a line that prevents intimacy.
The best thing to do to avoid facing an awkward situation where you don't know how to greet someone you've just met is to use English to converse. English just has 'you', which is free of all the intimacy and respect issues that Bangla contains. I had a private tutor, and for the five years he tutored me, I always addressed him in English, because I initially didn't know whether to address him apni or tumi, and it later became a habit. The problem with this, however, is that you cannot talk in English with everyone at all times. Sooner or later, you have to switch to Bangla, and the problem will arise again.
While at work, you're expected to greet your coworkers with apni, though there is no such writ that dictates so. Even when you become close to them or they are of your age or younger, you are expected to not cross the line between apni and tumi. Reason? You need to show respect. That is part of the professionalism-package that you should be selling in the corporate world. Personally, I think that is a polite thing to do, although it gets difficult when you work with your friends.
The same happens when you meet someone just a year or two senior to you. Do you call them bhaia/apu, or just by their names? And should the tone be tumi or apni? It's confusing, eh? At times like that, just ask them whether you should call them bhaia/apu or not. This solves the awkwardness, and chances are that they'll reply, “Call me whatever you want,” thus permitting you some freedom at that. However, if you're using the bhaia/apu suffix, it's better to go with apni. If you still want to go with tumi, make sure you tell them not to be offended.
Let's move on to the next case. You've just been introduced to someone your age, or someone you're academically equivalent to, or somebody who'll be joining your band. Either way, if the two of you have to share an elongated period in each others' company, I suggest starting with tumi and gradually shifting to tui over the weeks.
But the issues don't end at that. Sometimes you can't decide by a person's appearance whether or not they're senior to you. On the first day of my university life, I called someone by his name, later to learn that he's four years older to me. It was awkward, but we laughed it off by deciding that he looks much younger. If in a situation like this, well, I say take the risk and use humour to correct yourself later if need be.
Bangla is a beautiful language. While some might consider the apni-tumi-tui issue to be a social impediment, I say it's a boon which allows us different degrees of expressing respect and closeness. But if in a dilemma regarding which to use, remember to be respectful, or even better, trust your judgment.