I presume you work in a school and that's why you're going at half term? If you can avoid that week, it would make a big difference to price!
To answer your questions:
- Which sites shall I browse?
For your first trip, I would recommend a package. Skiing is easy enough to arrange when you've got the measure of if, but when it's all new there's a lot to be said for just turning up and having someone else do the thinking. Inghams or Crystal are probably the best ones to go with.
I have used skiline a few times - they are very good at sifting through all the package deals to help find something suitable for you.
In peak season, you'll struggle to sort anything less than a week, Most accommodation won't take a booking of less than that, and midweek travel is harder. That means you have 6 days on the slopes. That's enough to get you going and make some progress. Even if you could book for less than that, the trip will be over before you've started, and you won't achieve much.
- What resorts or areas in Europe do you recommend?
This is a 'how long is a piece of string' question - and all depends who you ask!
For your first trip, I would avoid the big French ski areas - mainly because they are all super-busy at half term and lift passes tend to be really expensive as they cover a huge area, which will be of no use to you on your first trip. So I'd go for a smaller resort, as you don't need huge KMs of slopes. I'm a big fan of Austria and Italy over France (have skied in France a few times for comparison though), but to go for whole countries is a bit of a sweeping generalisation! The main thing you'll want is a decent ski school and access to a good number of blue slopes.
It will also depend on your budget, how long you want to spend on a transfer coach, what kind of accommodation you want... lots of things. Hopefully you'll get a good number of points of view on this thread and you can build up a picture of what you might want to consider.
- How much do you budget for a trip, all in?
At half term, I don't know what you'd expect to pay, so will leave that to others!
You'll need to factor in the cost of the package, plus equipment hire (tour operators usually offer good rates for that), lift pass, ski lessons, lunches etc.
Other things to think about...
The main thing is what kind of accommodation you want. Your options are broadly:
Self catering (reasonably rare in a TO package) - the cheapest way to do it, but I would avoid it on your first trip. The last thing you'll want to do is cook, and eating out can feel like a bit of an undertaking (when I'm tired, the last thing I want to do is to put all my warm clothes back on to go out, trudge around looking for somewhere to go, hope they have space, etc etc)
Half board in a hotel - the most common option, and is exactly what it says. Huge variation within this (basic, unexciting but generally tasty food , right up to gourmet 7 course dinners!)
Chalet catering - more seen in France than elsewhere, and not as common as it used to be (thanks to Brexit), but typically smaller than a hotel and meals eaten more 'family style' - round a table with other guests. Chalets often have some shared living space too, so you have somewhere to relax. And there is tea and cake when you come off the slopes.
Skiing 'stuff' - you'll need some gear - trousers and jacket being the main ones, but also decent ski socks and some thermals/layers. Plus gloves and goggles.
I'd look on Vinted/ebay/marketplace for some of this. Or if you have friends who go, maybe they'll lend you some stuff. I have lent all my thermals to friends going before (but I'd be less inclined to lend things that might get lost like gloves or goggles).
The first time you wear ski boots is a shock to the system! If you can, it might be worth a trip to a snow dome - just to get the feel of the boots/skis etc. It will make your first morning less traumatic.
And finally - prepared to get addicted! We went 'just once' 14 years ago, with some friends, just to 'see what it's like'. And now we go pretty much every year. I am the world's biggest wuss, a huge scaredy cat, a total control freak and (when I started) not very sporty... but there's something about being out in the lovely fresh air, in the stunning scenery, with only one thing to think about that just makes for an amazing holiday. I absolutely love it!