1.So I live in the high desert, goat-thorn/prickly stuff galore. I rode fixed for the first year I got here after moving from NYC and literally got 1-2 flats a week so I got frustrated after trying EVERYTHING in my budget to fix that problem and sold my bike. Now I am looking into buying a mountain bike next week to take on adventures, I would like to run tubeless(I am so new to MTBing I just found out last week that tubeless was even a thing). I read people said they use sealant? Slime? I am used to the oversimplification of my fixed gear so what ALL do I need to buy run a tubeless set up?

2. What is a better way to go about getting a new MTB for a beginner, buying a "cheap" MTB off ebay/amazon and upgrading each part as needed or buying a great bike overall from a LBS? My budget is not that big so I need to keep $$$ in mind.

3. When buying a MTB what are the most important/MUST HAVE QUALITY parts to look at when buying? What is the most integral part of a mountain bike that I should make sure has the best quality part?

Any advice or points in the right directions are greatly appreciated! Thanks!

1. Stan's is the most popular sealant for tubeless. It is latex based and will dry out. How often you need to refresh it wl depend on weather. I would ask the locals what they use.
Tubeless can be frustrating if you are not using tubeless compatible rims and tires.
2. Buying a great bike is always better. But it will depend on your budget.
3. The hardest/most expensive parts to upgrade will be frame, fork, and wheels.

1. I don't know. We're kind of in the same boat here. Just upgraded. I went from 26" tubes to 29" tubeless and it's fantastic. Can't say for sure if it's the tubeless or the size.

2. I bought both my bikes from Craigslist, both a couple years old but barely used. Bought a $520 bike for $300 (over 40% off from a new) and learned to ride on it (rode for years). Great for what I paid and needed. Just bought a $2,000+ bike for $800. True, a LBS should be able to fit you up better for the ideal bike, but for the kind of deals on Craigslist I'll be going with a 'close enough for me' fit from here on out. Plus, if you have an idea of what feels right, you should be able to do the same.

3. Fork and frame. I don't know anything about wheels (but they are pricey).

A good frame and fork, however if you get a more basic fork you can usually upgrade it depending on your preferences and budget.
A good set of tires, because you will notice a difference in corners, climbing and rolling resistance (Kenda is my choice brand)
A good set of tools, preferably ones you can take riding. Having cheap tools is nice until you're stranded and a tool breaks or strips itself. Good tools make working on your bike easy and fun. Besides, if you end up on a group ride or tagging along with others you never know what they may or may not have (I've saved too many riding buddies to leave my tools at home).

I ride a used 2013 DB OD Comp HT for $500. I've changed the tires to Kenda. Ride in fire roads and some single track in LA area (basically always dry and plenty of goatheads!) I run with tubes for now, it's a PITA either way just read the Brian Lopes book, one author gave up on tubeless and runs tubes.

I would buy a nice used bike and decide if you need HT or FSR first. Don't buy anything with MSRP < $1k. I wouldn't trust it. My bike is a comp model and it has Avid Elixir 1 brakes (entry level) but they seem to work fine so far.