Only my opinion.....but the best pump gun on the market, for the money, in my opinion...is the Browning BPS Hunter model ...in a 12ga with a 28" barrel ...listing for around $ 650 ....but selling new in my area for around $550 or so ....
Many of us ...into the clay target games...Trap, Skeet or Sporting Clays ...got into it with pump guns to start with. A pump gun - does make your 2nd shell a little slower...but not so much slower that its a terrible idea / while it may not be optimum for competition ...you can certainly shoot a pump gun for any and all the clay target games.
Even though I've invested in a number of shotguns for competition...for lots of reasons....I still have the pair of BPS Hunter models..I purchased in the late 1970's...one in 12ga and one in a 20ga...and I killed many thousands of clays ...and shot lots of upland and waterfowl with that gun - and they've been thru 2 boys learning how to shoot ...and now a bunch of grandkids as well...and still going strong after ...500,000 shells or more at least thru the 12ga model ...
There are other guns out there ...that won't break the bank...where most of your "target grade Browning or Beretta O/U's will be $ 3,000 plus " ...and they are very good pedestrian level guns ..not fancy guns ...but more than a lot of budgets can handle.....so if you wanted to look at another gun, I'd put the Browning Silver hunter ...semi-auto on your short list as well. Again its a solid gun / that will last for many years...no frills...but sells new for around $ 900 or so ..in my area.
Browning - Winchester are the same company these days ...( and both are under FN ) so the Browning and Winchester shotgun offerings will often look identical.
Nothing wrong with the Rem 870's ....but to get to the same level of quality as the BPS Hunter ...you'll need to look at the 870 Wingmaster in my view. But I think the BPS is a better gun ....cast neutral (doesn't favor left of right hand shooters ), ejects out of the bottom, the safety is on top of the tang - makes shooting with gloves a lot easier, Browning Invector Plus screw in changeable chokes are well made, good durable gun !
Beretta makes a lot of guns - in semi-autos you'll find at least 25 models ..and nothing wrong with them either.
Remingtons technology on semi-autos is pretty old ...the Rem 1100's and 11-87's ...but might be worth a look if you find one. Some of the older guns - if you find a good one may be a great buy.
Benelli is a good company -- a subsidiary of Beretta...and making very good guns. Nova and Super Nova are their pump guns ...a gun like their Super Sport in a semi-auto is their "clays" gun ...but most of the Benelli semi-autos are going to get up into the $ 1,500 - $ 2,000 price levels pretty quickly ...and they are "Inertia operated" vs "gas operated" - so they give you a little more recoil ...but Benelli has a "comfort tech" system in some of their pump and semi-auto models that helps a lot ..but adds to price too...
Get a gun with changeable screw in chokes....probably a good pump gun to start with ....then spend a season or so ...and figure out if you want to get into this in a bigger way. Most any gun that has a 3" chamber should be enough gun ...in my opinion a gun with a 3 1/2" chamber might be needed for waterfowl ...but there are a lot of magnum hunting shells made in 3" these days as well...but make sure the gun you pick has a 3" chamber ( so you can shoot 2 3/4" or 3" shells ). Many "target guns" will only be available in 2 3/4" chambers...
and no matter what ...make sure you have fun with the process...and welcome to our side of the gun hobby ....