As someone who spends a lot of time thinking about hiking and mountaineering, I spend a lot of time thinking about gear purchases. When someone asks me for advice, I tend to write them a dissertation. Here's something I wrote up recently. It's just my opinion, but I hope it will be helpful to folks out there in the same situation as the friend I wrote this up for.
Q: I have some hiking gear, but I want to get into cold weather camping, and I need to upgrade my tent, backpack, and sleeping bag. I am on a budget, and want to know how to use my money most effectively.
A: I would highly recommend asking around to see what you can borrow or make
do with before making investments. One of my friends has a saying,
"buy nice or buy twice". It's worth it to get the good stuff, but you
want to know what features you like and dislike in a particular piece of gear before you spend
that kind of money. I'm always willing to loan out gear as long as you
promise to guard it with your life, and you can probably find others who will too.
I think a good sleeping bag is probably the first investment I'd recommend. With a good sleeping bag, you can be pretty comfortable in the winter even in a 3 season tent. Down is more expensive, but it lasts much longer. I have a synthetic bag and a down bag that I bought around the same time, probably 5-6 years ago, and the synthetic bag is pretty beat down and not nearly as warm, where the down bag is still in great condition. There are other considerations in the down vs synthetic debate that could fill up pages, but durability is something that directly impacts your budget.
As far as a pack, as long as you can fit what you need in it, and it fits you properly, you don't need to get too fancy. Usually the more expensive packs are just lighter, which is great, but probably not a priority in the budget. If you have a local store that specializes in outdoor gear, they probably have some experienced pack fitters, if you're buying something big there(like a down sleeping bag), they might be willing to help you fit your pack. A good pack can be great, but if you have one that is usable, I'd wait on upgrading.
A good tent is a worthwhile investment too. Unless you're going to be doing a ton of solo trips, I'd recommend a two person tent, as they usually work out to a good deal less weight per person, and give you more room to get changed, etc. That's a little bit of a personal preference issue. I'd recommend trying to borrow some different styles and sizes of tent before you buy to make sure you can fit in it comfortably. A 4 season tent will be a must have on serious mountaineering trips, but they are expensive, and can often be borrowed or rented. You can probably delay this purchase for a bit as well.