What squads can I beat with my new Palps lead?

All these toons that you're looking at are great investments.

Dooku is necessary for GeoTB, so having him geared and Zeta'd is worth it. He's overall a good toon so if you feel like G12ing him you'd get your gear's worth.
Sion is a great Sith tank that taunts when he applies the debuff Pain. He can apply it with his basic, aoe, and when enemies damage him. Super annoying and like I said earlier synergizes well with DN.
Tarkin has a Capital Ship which is useful for TB and TW; but he might be the least useful in other PvP places. Not bad, just not as useful as the other toons you're looking at.
Thrawn is definitely a must have: good kit, can be thrown into pretty much any squad and work well, and a Capital Ship.

As far as who to work on gearing, that's dependent on where you feel like focusing:
Dooku for far long-term with GeoTB and unlocking Padme.
Sion for short-term Arena gains but long-term Sith team.
Tarkin for short-term ships and long term TB/TW.
Thrawn for long-term PvP and all-around usefulness.

You say you're ready for Thrawn's return so possibly think about saving up gear for him while farming Sion's shards, maybe throw some gear into Sion as well while you're at it. Dooku could use G12 and a Zeta if you want, but that might be too much of an investment for you right now.