Normal, plain clothes are no problem. The only thing you should think of is not having to "revealing" clothes, so better avoid bare legs and bare shoulders (short sleeves are ok) - especially if you're going for a teaching, meeting a lama or attending a ritual. If it is a festive celebration of course you can choose more festive clothes, but I wouldn't say you have to if you don't want.
In the West (and in India) you usually take off your shoes when you enter the temple (and leave them outside), whereas they usually keep their shoes on in Tibet. I noticed this is changing also in Tibet, however, maybe because of influences from China or the exile Tibetans in India, so if you're going to Tibet yourself, just check how the others are doing with their shoes and then do the same. Take off your hat also.
Other tips, if you want to make an extra effort, might be (but don't be too afraid if you forget the details, your intentions are far more important!):
* stand up if the lama or teacher enters, keep standing until he (or she) is sitting. If you want to blend in, you stand bowing slightly forward, keeping your eyes lower than the lamas, and hold the hands palms facing each other in front of your chest.
* don't stretch your legs and show the soles of your feet towards the altar, lama or buddha pictures
* don't step over other people, sacred texts or other sacred things
* don't put texts or other sacred objects on the floor or on the sitting cushions
* if you're a buddhist or at least don't feel uncomfortable with it (you don't have to be a buddhist to do it!), you can make three prostrations in front of the altar at entering the shrine room. If you go for a teaching you similarly make three prostrations to the teacher after he/she has taken his/her seat at the beginning. Again, only if you feel comfortable with it.