Tyranny of Dragons has received mixed reviews, so it depends on who you ask - some people loved it, some hated it. The Beginner's Box is probably the best entry point and a good value for the money. It has a longish (64 page) adventure for levels 1-5 that is generally very well regarded plus a rules booklet. If your group is enjoying 5E after playing through that I would recommend transitioning to Princes of the Apocalypse from there.
I 100% agree with this view. I would not recommend Hoard of the Dragon Queen and Rise of Tiamat for a newbie DM (or player). They're not
bad adventures, but they certainly require careful consideration. There are some encounters which, frankly, are just a bit too difficult for the average party. I think this was due to the module being designed while the rules were still being developed. Also, the treasure and encounter variety is lackluster in some places. In my opinion, Rise of Tiamat is the better of the two simply for virtue of being released after the rest of the books were available. Unfortunately, to get to RoT, you have to slog through some fairly uninspired / difficult sections of HotDQ. An experienced DM can fairly easy "fix" this, but I wouldn't recommend it as your first experience.
The adventure in the Beginner's Box, by contrast, our group found to be outstanding. It's an excellent introduction to the rules, and a fun adventure in itself. And, for my money, Princes of the Apocalypse is the best official 5e adventure available so far. My group is currently running this, and enjoying it tremendously.
For conversions, I've found it easiest to translate the classic 1e/BECMI modules. We had a lot of fun with the Sinister Secret of Saltmarsh series, and we've also played conversions of the classic Keep on the Borderlands, Forest Oracle (yeah, yeah... laugh it up, grognards... we enjoyed it), Master of the Desert Nomads, Drums on Fire Mountain, and various others. Most old-school modules can be converted almost 1-for-1. A room with 6 orcs in 1e can be translated to a room with 6 orcs in 5e, and they're about the same level of threat (very-dangerous-but-beatable to 1st level characters, still pretty-dangerous to 6th level characters). This is different to converting 3e or 4e modules, where 6 orcs is absolutely-deadly at 1st level and a total-joke at 6th level.
Also, the 3e prevalence of magic items doesn't convert well to 5e. Opening up any copy of Dungeon magazine (or the Paizo stand-alone modules, such as the Gamemastery line), every random bugbear guard has a potion of cure moderate wounds and a +1 axe... because 3e had wealth-by-level guidelines, and careful balancing of numbers, and suchlike. Random NPC 5th level wizard has two wands, and a +2 stat buffing item, and a +1 ring, and four other potions and scrolls. If you're doing a conversion from 3e, you want to take a serious look at cutting down the number of magic items. Maybe pick 3 permanent items and a half-dozen consumables per adventure, and that's it. Everything else becomes non-magical.
