I'm an Aliexpress fan and was wondering if anyone could recommend a particular ESP8266 variant and vendor? I think it's time I dipped my toes in the water, so to speak.
I would urge you to consider the ESP32 instead. See here and here for some simple comparisons of why the ESP32 may be a better choice at this point.
That being said, I have used a couple of different ESP8266 variants for some fun and projects and, once you get the hang of some things, they do work. There is a very nice monograph/user guide that has been posted on this board that I think would be a big help if you want to start using them and also want to save yourself some aggravation.
The ESP32 is going to be a few bucks more, but I still think it is the better buy if you just want to learn and use it in some projects. I just received one of **broken link removed** a few days ago - so far it checks out. I also have one of these. I really have not done more than some testing and scanning and running a few examples and the like. The additional BLE capability is intriguing and since the demise of the 101, I have been looking for a go to board to continue to play with BLE. Right now as far as I can see, BLE software and documentation is a little underdeveloped - hopefully, that will change. But, even without considering the BLE part, I think the ESP32 is superior.
I second the Wemos. Make sure it's revision 2. The mini is about the size of a stamp and very powerful. The 4 meg of flash can be setup as a 3M drive and 1M for code and can easily serve up a complete website. I posted a link elsewhere to a very good document on the esp.