For a first ski holiday, unless you are going with someone who knows how things work I would always recommend going with a package company such as Crystal, Inghams or Nielsen.
Everything is taken care of (flights, transfers, hotels, ski hire, ski school etc) and there is someone on hand to make sure that you get to where you need to be on the first couple of days until you get the hang of things.
Half term will always be the most expensive week of the year - which is a bummer if that is the only week you can go. In 2024, half-term in the UK coincides with much of Germany, Scandinavia and France, affecting prices accordingly. Packaged ski holidays are almost always Saturday to Saturday - so seven nights away, with 6 days skiing. Don't worry about resorts being snow-sure - it will not be a problem in mid-February.
Price - I've looked at a few resorts in Austria that I'm familiar with, travelling with Crystal for w/c 10th February and staying half-board (i.e. breakfast and evening meal). Prices range from about £1,500 for good, basic accommodation up to over £2,300 per person for 4/5 star.
In addition, you'll need: Hired skis, boots and a helmet, which will cost about €100 for the week per person. A lift pass is €275-€325 each for the week, depending on the resort. Ski lessons cost about €200 each for group lessons for a week - private lessons can cost as much as €175-€225 for a two-hour lesson, but you can share this between two of you. Lunch on the mountain in Austria is about €25 per person (a bit less in Italy, but much more in France). Drinks are about €6.
Personally I would not choose Bulgaria at half term. It is cheaper than Austria/Italy/France, but for a reason. A better budget option might be Andorra, or possibly Kranjska Gora in Slovenia just over the border from Austria (but not anywhere else in Slovenia), but otherwise stick to those 3 countries.