Because Anning’s damage is so low, DPS Anning really wants optimal supports: two Castorias and a Tamamo (for damage), a Reines (for stability and NP gain), or an Oberon (for raw charge). Anning refunds about 60% off three enemies with double Castoria, so for farming she’ll need a starting charge CE, and for CQs she’ll need cards to loop. She’s pretty straightforward in both cases, though—your usual Arts strategies will work for her provided she hits damage thresholds.
I’m not going to bother going into detail about lower-tier DPS setups for Anning since her damage is too low to make anything sub-optimal particularly viable, but if you want to try it, there’s a plethora of alternative Arts supports available. Sherlock and Lanling can play into her crit tools (although they don’t benefit fully from her buffs), Mozart and Paracelsus are generally nice budget supports for any Arts DPS, et cetera. You can just play around with whatever you have until you find a setup you like.
Support Anning is somewhat more interesting. She’s only worthwhile if she’s supporting non-Male Arts Servants, for obvious reasons, but even in those cases she has to compete with Paracelsus and Mozart. She sort of splits the difference between the two, bringing more consistent NP gain than Mozart (but less than Para), and more damage buffs than Para (but less than Mozart). In most cases you’ll want Mozart or Para for their specializations, but if you need something between the two—or really want crit damage buffs, NP charge, or buff block—Anning can stand out.
Support Anning will generally be best if she can drop her stuff and then go away. A taunt CE makes this pretty easy if Anning doesn’t need to NP. If you need the buff block, though, you can accomplish something similar using Chen Gong to explode Anning after her NP goes off. Anning’s CDs are short enough that you can keep her around, but that’s generally going to be less effective than replacing her with a fresh support asap.
Anning’s CE picks are pretty simple. In CQs she wants the Black Grail to offset her damage issues. For farming she wants Ocean Flyer or Painting Summer, since she needs the extra charge. As a support she either wants Kaleid for an instant NP or one of Outrage and GudaGuda Poster Girl to get rid of her quickly.
Utility and star gen CCs are good picks for Anning. Star gen CCs can bring Anning closer to flooding the field with her second skill, and utility skills (like party NP damage buffs, heals, and buff removal) make Anning a better support.