Just my opinion:
Plan to use the Buhlmann ZHL-16B or C (close enough to the same to not matter for just starting tech diving) algorithm with Gradient Factors until you've gotten enough education on deco theory and algorithms to make your own informed choice to use something different. Or stay with Buhlmann+GF, as many, even experts ,do.
Just about every reputable trimix-capable computer on the market supports Buhlmann+GF. There are a couple of exceptions and those that don't seem to be less popular (and maybe that's one reason why).
You're in Canada, I gather. In that case you should have the option to buy a Heinrichs Weikamp computer, which are not sold in the U.S.. So, I would look at:
Shearwater Petrel 2 or Perdix
HW OSTC 3 or 4 (the 3 is less expensive than the Shearwaters and completely equal on specs. The 4 is more expensive and even nicer)
Ratio iX3M+ Tech
Ratio iDive Tech
Divesoft Freedom
Liquivision Xeo
Of them all, I like the OSTC 4 the best (based on specs, reviews, reading forums - I have not held one). If it were available in the U.S. and had AI I would have already bought one.
They're all basically brick-like form factor on your arm except for the Ratio iDive which is a wristwatch form factor.
If you want something that will do the job and you're on a serious budget, you could also look at:
Hollis TX-1 (~$450 - puck)
Dive Rite Nitek Q (~$400 - brick)
Personally, I would wait another couple of weeks, though, to see what is announced at DEMA. If ScubaPro finally releases their version of what was the SeaBear H3, that is definitely another option to consider. I had a Petrel 2 and an H3. I love the H3, sold the Petrel and have not missed it. I also have the Hollis TX-1 as a backup. It's fine (for me, anyway) as a backup, but I would not choose it as my primary. For my primary, I want something that is more readable in really low viz conditions. The H3 is every bit as readable as the Petrel 2 was, and it's a lot smaller form factor (wristwatch sized).
There's not really "integration" with your desktop computer. It's more a matter of running software on your desktop/laptop or tablet that uses the same algorithm as your dive computer, so you can plan it on your desktop or tablet and then expect your dive computer to work the same way when you actually do the dive.
For desktop/laptop, there is Multi-Deco ($$, Windows or Mac), Subsurface (Free, Windows, Mac, or Linux) and others. For tablet use, there is still Multi-Deco, PastoDeco (on Android), and others. I use Multi-Deco on Windows and Mac, Subsurface on Windows and Linux, and PastoDeco on Android and I like them all.