You will get what you pay for with a $300 budget. Shooting clays, that would mean - between targets and ammo - that you will have spent more than your gun in about 30 rounds of trap or skeet, even less with sporting clays.
While a typical Mossberg or Remington pump is not a bad gun neither are ideal for clay games where doubles are part of the routine. It can be done, just not as effectively, resulting in lower scores than if a semi or O/U had been used. Unfortunately, with your limited current budget, those are not part of the possibilities.
IF $300 is the MAX you can go, then get whichever pump feels good to you and go have fun, realizing that if you really want to improve and get into competition, etc., a different gun will be needed shortly.
Trap singles and skeet are completely different from each other, especially when it comes to gun styles. As you begin this path of smoking clays, be prepared to want to upgrade through different guns, lessons, and lots of practice.