Do you have a good sense of what you should be looking for in a fit?

There are various videos and guides. Worth some watching and reading.

My brief summary:
- Saddle height: when pedal is at lowest position, your foot should be roughly flat (some people have the toe down, some have the heel down, but either way the deviation from horizontal shouldn't be too much, like max 20 degrees or so). Your knee should be flexed a little (20 to 30 degrees). There should be enough adjustment left in the seatpost to let you vary the saddle height from this position by an inch or so either way.
- Torso angle: with hands on brake hoods and arms pretty straight, your torso should be around 45 degrees from horizontal. More aggressive fits will have this a little lower.
- Reach: with hands on brake hoods and arms pretty straight, the angle between your torso and your upper arms should be around 90 degrees. Then lower yourself until your forearms are horizontal and pedal; your knees should not overlap your elbows by more than an inch or so.
- Standover: with both feet flat on the ground, you should be able to stand over the top tube with a bit of clearance between top tube and crotch. This isn't strictly necessary, but it is convenient to be able to stand over your bike without contact.
- Balance: place hands on brake hoods, then lift them off; you should be able to hold yourself in that position without exerting your back/core muscles much, meaning you should be able to hold that position for a minute without difficulty. If you have weak core muscles this might not work, but most reasonably fit people should be able to do it.

The above describes a typical road bike fit for the typical rider.