Boy Howdy! What a question! That's kinda like asking a group of 20 golfers what their recomended driver is. Youre gonna get 23 different answers cause some folks got more than one favorite.
IMHO for your intended purpose-
If you don't need more than 2 shots before reloading for the next series of targets I'd go with an over and under arrangement. They are simple and efficient. They can be unloaded and reloaded quickly.
You should look for something in a 26 to 28 inch vent ribbed barrel. Why vent ribbed? Well of course there is the stability and weight issue but also as an older shooter myself, my eye sight ain't what it used to be and you can snap a Hi-Vis sight on the front of that thing with no tools.
To lighten up the recoil a tad I'd go with a 16 or 20 guage shotgun. You will have to be a better marksman with these rather than a 12 guage because you will be putting less shot in the air as the guage increases but these are a good compromise. Here's a fairly good article on guage and shot load size- http://www.chuckhawks.com/intro_gauges.htm
The location of the safety is a consideration. The Mossberg safety is behind the receiver and typiclaly de-activated with the thumb while the safety on the Remington is located on the trigger guard and de-activated with the pad of the trigger finger without shifting the grip. Most all shotgun mfgrs. will locate their safety switches in one of these two locations. I personaly prefer the Remington style so I shoot a Remington 870 pump.
Now, getting everything you want and just the way you want it for less than $400? Well, you are probably going to have to stay away from most gunshops or even Wal-Mart. I'd be looking on Gunbroker.com or other such firearms auction sites. Another great place to pick up inexpensive but good firearms is at local estate auctions.
A good example of what you might be looking for would be a Remington Model SPR310S Sporting Over Under 20Ga Shotgun. It has a satan finished walnut stock, nickel receiver, mid and front bead sights, 28 1/2"Bbls with an overall length of about 47" and weighing in at about 7lbs. With a 3" Chamber, rubber recoil pad, chrome-Lined sore, vented rib, and a single selective "Gold" trigger, this firearm can be had for between $350 and $450.
For me (in a perfect world) it would be a Remington Premier Field Grade model Over & Under with a 28 inch barrel chambered in 20 guage Unfortunately, I'd have to learn to live with a tang safety switch though. But, in a perfect world I could adapt.
I probably raised more questions for you but hopefully I gave you a little food for thought.
But, then again, what do I know? I shoot trap with a Remington 870 12 guage with an 18 inch barrel that is set up for defensive purposes- because I'm too cheap to buy a specialty shotgun and just because I can.
Actually I am a certified instructor in pistol, rifle, shotgun and personal protection but my first love is and always has been pistol.
Good luck with your decision and I hope you enjoy your new sport.