I can only really answer from my own perspective, and I only wrote real code (other than simple BASIC programs) on an Atari over the last couple of years.
In terms of classic tools IDEs were relatively light compared to tools you get today. I worked using Devpac 3, which is a good assembler that includes some useful features and a debugger. STOS could be regarded as a BASIC IDE specifically for games, and GFA BASIC offers pretty much everything you need form a coding/debugging point of view in one place.
BASIC and/or 68k Assembly. I'm sure there were/are some games written in C, but most people seem to prefer the lower barrier of entry offered by BASIC, and from there tend to head full-on into assembly language.
Useful Resources Today
Atari Forum's WIKI has some useful documents and tutorials, and the #atariscne IRC channel on Freenode has some very smart people in it!
Checkout the Dead Hackers Society's Coding files section for tools and libaries to get you started.
DML (aka Douglas Little) recently published AGT (Atari Game Tools); a suite of graphics tools and a C based engine for quickly prototyping games on the Atari STE. The wiki includes some tutorials and you'll have sprites on screen in no time.
Running The Programs
If you develop on the Atari then you'll be building an Atari executable file so you'll be able to simply launch it with a double click. I personally wrote some C and assembly on my Atari, but I ran into issues with screen mode switching and using the debugger for my game (I was working on a Falcon in 256 colour mode). Later on in my project I wrote the code in Devpac but running under the Hatari emulator, that meant I could use Hatari's debugger to work through issues.