Vatican tickets

The Basilica is free, except for specific places inside it. The Museums require tickets, which you should buy in advance to avoid long lines.

Backing up Zoe here, the only part of Vatican City which does not require a ticket and/or tour is the larger part of St. Peter's basilica. Other pieces of the church - such as the dome - do have fees.

I have found where you can purchase tickets for tours, early entry or
skipping lines

I assume you've referenced the Vatican's website for your information?

http://mv.vatican.va/3_EN/pages/MV_Home.html

You can only access certain buildings of the Vatican, not all areas are open to the public. The main ones are the museums, the Basilica, the square and some specific tours that must be booked in advance. For the museums and tours you need tickets, for the basilica you don't need a ticket but must wait in the security line to enter. The square is free and requires no waiting in line or security check

Donna

When people say "the Vatican," they're usually thinking of St. Peter's Basilica and the Sistine Chapel. You've already been told about St. Peter's, but just to clarify, the Sistine Chapel is inside the Vatican Museums; there is no way to see it without buying a Vatican Museums ticket or tour, and then trooping all the way through the museum (with several thousand other people, unless you pay more for one of the early or late tours, like the Pristine Sistine tour offered by Walks of Italy).

One place on St. Peter's Square that you can get inside without a ticket, security check, or wait, is the post office. Famously, it's more reliable and faster than the Italian postal service.

Other buildings in Vatican City are not open to the public; you'll see the Swiss Guards outside them, and unless you have official business in them, the guards won't let you in.

There are tours of the gardens in Vatican City - not cheap, but if you have an interest you may want to do this: http://mv.vatican.va/3_EN/pages/z-Info/MV_Info_Servizi_Visite.html#giardini. Otherwise, these are not a place you can visit on your own.