I'm running the Lost Mine of Phandelver adventure for my son and his friends as DM. With the future in mind, I want to get more published adventures. How can I find out a comprehensive list of all the published adventures for 5e, ideally with enough information (character levels, etc.) to determine which I might want to purchase and use?
Princes of the Apocalypse looks like your best option. It is set close to Phandelver, has the same look and feel (same cartographer) and is for characters level 1-15. It really starts at level 3 though.
Your other options are Hoard of the Dragon Queen (1-7) and Rise of Tiamat (7-15) A lot of work required by the DM, but both nice adventures.
You could also try the Wizard's Amulet by Kobold games for a free 5e adventure https://froggodgames.com/sites/default/files/promos/The%20Wizards%20Amulet%20(5E).pdf
The best way to get a comprehensive listing of adventures for D&D, or any system, is via RPGGeek, a comprehensive listing site. In this case it's a little tricky though.
- Go to the D&D 5e page and filter the linked items by Category: Scenario/Adventure/Module. That gets you all the official WotC adventures.
- Then, go to the 5e Compatible page and do the same to get all the third party published adventures.
Unfortunately you have to drill down on each item to see level information (and even that's dependent on what they thought to put in the description). See Is there a good searchable database of D&D adventures? for the previous question on this (tl;dr - no good solution exists).
The best way would be to visit my site, where I keep a list of all the adventures published so far for D&D 5E. I maintain two lists - one by publisher, and one by level.