Questions to ask as I shop for a tonearm?

My advice for any newbie is to buy a complete turntable... And save what may become a long, expensive and painful lesson in what can go wrong buying a separate tonearm.  
For folks with experience, or who have a mentor, great. Without either... Yikes!                           
So I say stick to a ready made pairing of a great table which comes WITH THE ARM.                          
Thank me later.        
Good luck.                  
PS I have been at this a long long time, and I would not consider buying a separate arm unless it was like one of the very best, Kuzma 4 point etc.. And that mainly because they only come as a separate purchase a lot of the time. Anything cheaper.. you can get a good ready made together.And if I did, the arm would be added by a pro.. Not some tinker.

The main function of a tonearm is to register the cartridge in the record groove, and to maintain this registration as the stylus transits the record.

This is accomplished by allowing fine adjustments to be made, and having those adjustments remain stable. Resonance is also an issue.

Enter Trans-Fi Terminator air bearing tonearm, a superb tonearm within your price range. I own two of them, both on high end turntables, one a commercial product, one a DIY air bearing. They are highly adjustable, stable, and exceedingly "tweakable". I have had good results with a light MM and with a higher end Koetsu, just by changing the tonearm mass with weights and adding damping.

The only problem is that the tonearm does not work with a suspended turntable, because the beam of the tonearm must be absolutely horizontal for the air bearing to work. Since neither of my TT is suspended (they sit on a concrete foundation bedded in rock far from heavy equipment), the Trans-Fi is ideal for me.

Good luck!

I've had a Grace 707 arm of 20 years and sold it; Maplenoll Ariadne air bearing arm/ turntable--the best I've had for sound up to now;  VPI Scout w/ arm; VPI Classic w/ arm; Avid Diva II w/ Origin Live Silver II arm; Super Lenco TT w/ OL Silver II arm; And then the same Lenco TT w/ Trans-Fi Terminator arm with all the upgrades.  The Lenco w/ Terminator was the overall second best set-up.  It did have the best dynamics and clarity of all of them.  I'll second the Trans-Fi Terminator for overall sound and use.  VERY TOUGH to find a TFT on the used market.  Mine was $1275 with everything including all parts of the air pump and delivery system.  I was hoping to at least equal the Maplenoll TT/ arm in sound and have it be less potentially dangerous to the cantilever in the air went out.  NEVER a problem with the TFT as it requires very little air to float the arm.

May have one for sale in the very near future.  I had to spend quite a bit more to beat the TFT.

Bob

@dorkwad TFT is back on sale apparently.  I think it may be too much for the OP to deal with.
@elizabeth The OP inherited the table and it's pretty decent except it comes with weak Rega 250 arm which s/he wants to change out. 

Check out the Jelco line of tonearms as they fall right in your budget and offer performance above your budget.