Classical - If you are a classic person, then start by considering classical composers like Pachabel, Bach, Beethoven - the typical songs being Canon in D, Trumpet Voluntary, Wedding March, etc. These can be easily played by a string quartet, harp, pianist or organist. A handful of churches will only allow for sacred music, so be sure to ask your officiant if you are getting married in a church. And most churches will have a pianist or organist from the church that you either have to use, or will have the option to use. They can typically play most of the classical music.
Modern - If you want to bring a twist to your ceremony and you have the option to do so - like at an outside venue, museum, etc. - then you can consider almost anything and everything you love. You could play Bruno Mars, Amazing, as you walk down the aisle and the Beatles, All You Need is Love, as you exit, anything goes! Get creative, think outside the box. Play the songs you love! Think about taking a popular song and having it played on a classical instrument like a harp or acoustic guitar. Or if you have a couple of first dance options and only have to choose one, then think about using the others during your ceremony.
Depending on what instruments you have at your ceremony, I would recommend seeing if they can also play for your cocktail hour. Or you can also have your DJ or band play for your cocktail hour as well. It usually makes the most sense to have someone that is already going to be there to play during that hour. Unless you want to bring in a specialty instrument like a steel drum or an accordion! The cocktail hour is also a place to have fun with music. If you are Irish and you want to bring in Irish music into your day, this is a great place to do that and wow your guests. It is also a good place to play music that you may want to hear, but is not really right for dinner or dancing music. Think about it, it could be the perfect touch that your wedding needed!