Eric, first of all, welcome to Sailnet. You'll find this is a great place to get information, or just "hang out." Just don't ask any stupid questions and you'll be fine.
Invest in the Maptech Embassy Guide for Long Island Sound for your trip out there, and even while you're in Mystic. It's $45, and it has tremendous information on all the harbors between NY and Block Island, including Mystic of course. Well worth it in my view.
As far as I know, there really aren't rental moorings far enough up the Mystic River dinghyable (word?) to town. There are several marinas south of the bascule bridge that are less expensive than the Seaport (by the way, if you join as a member of the Seaport you get a fairly deep discount on dockage, and it's actually kinda cool to be docked there, particularly after hours when the grounds are closed to everyone except boats docked there). You could try Mystic Shipyard (860-536-4882), Brewer Yacht Yard (860-536-2293), or Mystic Downtown Marina (860-572-5942). These each are nice, and each are below the bridge, which actually is a very nice thing, as holding station waiting for that bridge really sucks. There are a few other smaller marinas in the area too. Oddly enough, you will see tons of boats, but there are limited facilities for transients.
In terms of anchoring, there really isn't much up the river. I guess it depends on the type of dinghy you have and how far is acceptable to you. If your dink is an 11' Boston Whaler with a 25hp engine, you can anchor or grab a mooring at the base of the river and be able to run up to town. If you're put-putting around in an inflatable with a 4hp motor, you are going to have a hard time finding an anchorage close to town, with one possible exception. If you have a shoal draft vessel you can travel past the Seaport, and anchor to the north of it. If you can get in there and find space, it's actually a great spot. The problem is the depth, and that there isn't much swing room at all (you may want to consider two anchors). I've never anchored there, but there always are boats there when we go.
Enjoy, as Mystic is a great spot, notwithstanding some of the challenges in getting there (all of which are manageable with some planning).