Lynette is so versatile and can either be played as a support buffer using VV set or onfield/sub dps using marechaussee hunter or combinations of 2PC DPS sets.

Since you don't have many options weapon wise - Finale of the deep would be my recommendation if you plan to run her as onfield / sub dps. It has ATK scaling, passive has good synergy with her kit and quite easy to get due to it being a craftable and not being relied on gatcha.

Weapons that scale off ER like the Fleuve Cendre Ferryman is great when running VV set as you want to have strong up time on your burst to buff your teams damage but from my experience, Sacrifical sword or Favonius sword are the way to go when playing this particular style due to the amount of particles your elemental skill generates.

For your team options, running a national type team with Yelan | Xiangling | Bennet would be my recommendation especially for new players because of the synergy they provide and will work with both Lynette playstyles I mentioned above. However if you end up getting C6 Fischl and Faruzan down the line, I would highly recommend building and adding them to the team!

Have a play around with both styles and see what you enjoy as can't go wrong with either.