Hearthpwn

In Hearthstone combos are fun, and often strong enough to help you win, and winning is fun.

If I were new, I would focus on the Classic Neutral Epics that will never expire and will help as much or more than any legendary.

Wide Decks: Sea Giant

Hand Decks: Mountain Giant, Doomsayer

Murloc Decks: Murloc Warleader

Then switch to the Classic Neutral Rares that will never expire and also enable certain deck archetypes.

Wide Decks: Knife Juggler, Defender of Argus

Hand Decks: Wild Pyromancer, Twilight Drake

Murloc Decks: Murloc Tidecaller, Coldlight Seer

It would take quite a while for a new free-to-play player to craft two copies of all ten cards.  But they would have a strong and flexible foundation to play with forever.

Great advice already.

I will add that budget decks tend to have two general themes: wide or hand.

A wide deck tries to play many minions for as little mana as possible to overwhelm the opponent quickly.  As others have said, Hunter and Warlock can do this well with Classic cards for new players and budget players.  If you enjoy playing wide then save your dust to craft two copies of the epic card "Sea Giant" which is foundational to any wide deck.

A hand deck tries to stall the game until its selection of large minions becomes unstoppable, or until some clever combination of cards is in hand for some kind of amazing combo play.  Priest does this best with Classic cards for new players and budget players.  If you enjoy playing hand then save your dust to craft two copies of the epic card "Mountain Giant" which is foundational to any hand deck.

When you play with a friend you can borrow one of their decks.  Find people with a more complete collection and use friendly challenges with their decks to learn what you find fun without spending a dime.  (I am davidvs#1835)

Yes, spend gold to get 10 Classic packs, and then 10 packs of each of the 2019 expansions.  That will guarantee you one legendary card from each of those four sets.  Most legendaries only add 2-3% to the win rate of a deck.  They are not crucial.  But they can be fun!