Most classes play just fine with stock weapons. You can play around with the different unlocks and see what you like. Most of these (non-vintage) can be had pretty much for free is you ask.
Going in order, I'd probably recommend:
Scout: Stock Weapons
Soldier: The Equalizer is nice (when you take a lot of damage your speed increases)
Pyro: The Axtenguisher (light someone up, axe them for insta-kill)
Demo Stock Weapons
Heavy: Sandwich (portable health kit for you or your medic)
Engineer: Stock Weapons
Medic: For me, this character is in most need of unlock weapons: the kritzkrieg for defense and the Ubersaw are must haves.
Sniper: Opinions vary on this. Stock weapons are ok. I like Jarate + Bushwacka, some like the Razorback (blocks spy backstab) or there's the Danger Sheild (+25 health).
That doesn't mean you won't have fun playing around with most of the weapons. The Demo Highlander + Charging Targe makes playing the Demo a whole 'nother experience. The Gunslinger helps the engineer be a more mobile class for fast-moving offensive lines. The Sandman can stop an ubered medic in his tracks.
If you don't have a large selection of unlock weapons, when you're in-game feel free to politely ask if anyone has any spares to...spare. Usually someone will donate a weapon or two for a new player