If you visit the museums on your own, you go at your own pace. Either buy the Vatican guidebook or rent the audio guide if you want more information. The objects all have labels, but with minimal information (name, artist, date . . . ). If you rent the audio guide you will have to backtrack a long way to the beginning of the museum to return it and retrieve the photo ID deposit. Do not underestimate how big the museums are. If you do it on your own, it's worth the extra money to reserve tickets in advance and avoid a possible very long line at the museum entrance security.
If you just want some highlights, a guided tour may be worthwhile. If you want an in-depth tour, a (pricey) private guide is what you'll want. Tours can exit the Sistine Chapel directly to the church, a very short walk and no security check. If you go on your own, you may not be able to take advantage of that exit. Some people report having an easy time doing it, others say they weren't allowed to. If you're on your own, you'll have a long walk through more (less interesting) parts of the museum and then a long walk to the church and line up for the security check.
You can visit the church very early in the day, when the security line is short (or non-existent) and the building is relatively empty and quiet. If you want to climb the dome, you should plan to do that when it opens (10 am maybe?). The lines can get pretty long. Space is limited and it looked to me like they were only letting people in at the rate others were leaving. It may thin out late in the day, I was there in the morning.