You'll find lots of advice around on what to wear for competitions, but its also important to dress appropriately for lessons and practice sessions. Wearing clothes that might hinder your dancing during lessons and practices will keep you from getting the most out of this time.
The most important thing is proper foot gear. Getting a good pair of dance shoes should be high on your priority list once you've decided that you're serious about ballroom dancing. Don't be fooled by the fact that ballroom shoes resemble normal dress shoes - they are more flexible, fit more snugly, and have suede soles. Also, ladies may be the ones wearing high heels, but proper dance shoes are just as important for the gentlemen.
In a perfect world everyone would go out and buy shoes as soon as they started dancing. In reality, you might have to make do with something less ideal for a few weeks. In this case, the first thing you should know is never wear sneakers for dancing. Sneakers and other shoes with non-skid soles won't allow your feet to move smoothly over the floor, and dancing in them might even lead to joint injury from the shoes causing your feet to stop too suddenly. You should also avoid any shoe that is very heavy and you should not wear anything with an open toe for standard or smooth. Your best bet is a shoe that fits comfortably and can be drawn smoothly across the floor; a flexible sole is a good bonus. It may be worthwhile early on to locate a pair of 'substitute' dance shoes that have some of these qualities and one of slicker varieties of rubber sole - you will continue to find these useful for certain practice areas and clubs that may have more dirt on the floor than you would like to get on your expensive dance shoes - (ccs)
You should also pay attention to your clothes when dressing for a lesson or practice. Jeans can be a problem - many styles restrict movement. Heavy or baggy tops will interfere with taking a proper hold. In standard or smooth you should also be on the lookout for anything on the front of your clothing that could potentially catch onto the front of your partner's clothing; you'll find that experienced ladies practice in shirts or trousers with a side closure or elasticised waists because a front closure might catch on their partner's fly. Ladies skirts should be full enough so that both the lady and gentleman can move freely - some skirts that seem full enough to the lady when she's alone aren't full enough when the gentleman needs to step between the lady's legs. You should also keep in mind that what is good for one style might not work well for another. For example the long, full skirts that are ideal for practicing standard in don't work as well for Latin - you spin and three bars later your skirt finishes spinning!
If nothing else suggests itself, dancers of either gender will probably feel comfortable showing up at most classes in a decent pair of slacks and a comfortable top, either short or long sleeved as appropriate to the season. For technique-oriented classes it can be helpful to have a shirt that is a little closer fitting at the waist than you might normally wear, as this will make it easier to gauge your progress in learning the proper body motions - (ccs)