I'd recommend that you start by reading the book Lighten Up. It's a great read that will help direct your thinking as you start on this path.
Next, I'd look hard at spending your money once. By that I mean buying lightweight gear the first time, rather than buying cheaper heavyweight gear that you'll want to replace later on. If you start with an ultralight mindset and gear, I think you'll enjoy the sport much more.
The big three are tent, backpack, and sleeping bag. Starting with a light tent is great. The Double Rainbow is great, as is the Scarp and the Rainshadow. Your Tarptent thinking is very good.
Renting packs is good because you'll likely carry too much gear the first few times as you whittle down your load.Later on, you can do packs (Golite is a good place to start) once you have your loads to a size you're comfortable with.
Any new items you'll be buying, do some research and get the lightest usable items. Often the ultralight stuff is cheaper anyway. For example, I would highly recommend a Trail Designs Caldera Cone stove and pot. Alchohol stove- great choice. You could also do a canister stove such as those by MSR or Primus. Easier but heavier.
So pick carefully- from your rain jackets to your boots (get running shoes instead) to your cooking pots (one needed- titanium). The more you can select carefully and minimally, the easier the walking will be. Sadly, Alps offers few ultralight choices.
The most expensive ultralight items are sleeping bags. High quality fabrics, down and synthetic insulations are expensive. This will likely be your biggest investments. Having a lightweight and warm bag that compresses small is a really important part of a lightweight system.
Lots of great reviews on this site- you'll probably save yourself a lot of money in gear by getting a membership. Sure- I'm staff now, but I was a fan far before that ever happened.
Best of luck- have a good time!