I would like to try room calibration for my poorly designed half assed HT room to see if I can obtain simple improvements. Don't know the first thing about it though. Could someone suggest a microphone and simple to operate software package?

I've been trying out Trevor's recommendation of AudioTools from Studio Six Digital (for iPad/iPhone). So far it's been pretty good using the iPad's internal mic. The base app needs a bunch of plug-ins depending on what you want to do. Or you can buy AudioToolsMax which comes with all the plug-ins enabled. Either way it can be a bit pricey. Also, It's best with older iPads with a headphone jack otherwise you'd need to buy a decent line out for it (or use bluetooth). Given that ...

I've also tried REW, which is free, but never took it too far. Amir at Audio Science Review has a decent write up on it: room-measurement-tutorial-for-dummies-part-1

Ditto.
I'm using the Audiotools LARSA with an older tablet. I don't trust the quality if the mic very much but you can find ways to get better quality mics for the testing.
Some screen images of my room sweeps with the program can be viewed in this thread:
https://www.axiomaudio.com/boards/ubbthreads.php/topics/435113/all/starting-to-oggle-new-gear