I bought the complete pack over the weekend after hearing about these games for years and now I'm not sure where to start.

I almost always prefer to play a game series in the order of creation but when I watched some gameplay of Mean Streets those flight-sim sections looked really rough... and I figured I'd be playing through the story again once I got to Overseer anyway so skipped ahead to that. Only to find out the game is set up as and would probably be executed throughout like a prequel. Complete with a framing device involving characters I'd never seen.
So... Under a Killing Moon then? Or Martian Memorandum?

Mean Streets.

The Tex Murphy series as others will tell you is largely self-contained where you CAN play the games in any order. However, that doesn't mean you SHOULD.

I may be a little biased since my introduction to Tex Murphy started with my dad bringing home a copy of Mean Streets (and some futuristic bounty hunting thing), but anyway...

Mean Streets is unique in that it does have the "flight simulator" and the action bits (adjustable difficulty is a good thing), but it really is the introduction to the series.

Access did a very good job in building the Tex Murphy universe from one game to another so that, while a new play CAN jump in any where, only those who've followed the series from its inception are really able to get all of the references and call backs to previous games the first time through.

I'd really suggest you stick with Mean Streets for not only that reason, but just how many innovations they had in the game.

RealSound was revolutionary (digitized speech from just a PC speaker), as well as the on-call secretary (v for Vanessa in the speeder) and information broker.

You don't have to mess with the "flight sim" interface if you don't wish too. Navigating the speeder is as simple as pressing N, entering a nav code and pressing A for autopilot.

Start with Under a Killing Moon.

yes don't bother with the first 2, start with UAKM