Considering you're in texas, a co2 gun could suit your needs nicely all year round as a low entry cost option.
A pump gun just gets old and annoying after a few shots and the action is very slow and noisy, so for hunting it can be a real PITA, or a gentleman's challenge.
Single stroke pneumatics are generally only suited for paper.
Springers have a steep learning curve to be used effectively, but aren't too loud or slow to operate. They are also of course self contained and rarely need any maintenance.
The discovery is a great starter option for minimal target shooting (using the bicycle pump gets old) and loud hunting. They can easily be tuned and customized between calibers for anything from rat to fox size game. With some more in depth mods and a .25cal barrel, you could use them on coyote at close range.
An upgrade of the discovery is the marauder. The main point is being integrally suppressed.
The crosman guns have an amazing aftermarket. They're like AR15s, the sky is the limit.
Please for the love of god do not buy another gamo product, or anything branded by a gunmaker, e.g. ruger or remington. All junk.
For a spring gun, look at RWS/Diana, and Weirhrauch. Make damn certain any optics you put on a springer are designed for it. I've seen scopes that have spent years on magnum centerfires get destroyed by springers in minutes. They don't have to be expensive, just designed for that use. Pumps and tank guns can use regular optics.