Welcome to the forum! Jig and Texas rig fishing are almost essential skills. There are a few things that can heavily influence the feel. A sensitive fast or extra fast action rod is helpful, as are low stretch lines like braid and the appropriate size/shaped weight for the depth and he cover you're fishing. There is also a big difference between say, an inexpensive Wal-Mart rod and a gen 2 Mojo Bass or Tatula, Aetos, etc. and that better stick will most definitely have a lighter feel with a more responsive blank, but....
the technique to fishing bottom contact baits is much, much more important and once you get the technique down, you'll appreciate the better tools much, much more. I (like many other forum members, I'm sure) learned to fish jigs and T-rigs with old fiberglass and cheap slower actioned graphite. The rod you're working while not a Cadillac, is still much better than the Ugly Sticks lots of us used and can make a decent rod to learn on (it being shorter than 7'-7'3 rods might sacrifice some casting distance, but it'll make accurate, close presentations and skipping much, much easier).
When fishing a jig, C-rig, T-rig, drop shot, or any other bottom contact bait, you need to remember that it's a bottom contact bait. I've spent a lot of time over the past few years teaching friends and family members to fish and the biggest mistake I see them (and tons of people with lots of experience get lazy/distracted and do this, too) making is relying on the reel to move the bait. When fishing these techniques, the general rule is that you should be moving your bait with the rod, and not the reel. There are exceptions, but this is the general rule. The reel is essentially just there to collect line and reel down quickly once a fish strikes to remove slack before the hook set. With the bait on the bottom (and I suggest learning with something like 3/8 oz), lift the rod tip slowly, allowing the jig to bump into the gravel, sticks, roots, tires, or whatever is down there. You'll feel every time that jig hits something. Over time, you'll learn to differentiate a soft bottom from gravel, wood, weeds, etc., it just takes patience and a little practice.
Another mistake I've seen and have been guilty of is losing a good connection to the bait (which will mess up your sensitivity badly) and it can happen a few ways. First, a relaxed palmed grip on the reel will allow you to feel a lot better than a death grip. This isn't typically a fast moving bait with resistance like a spinnerbait or crankbait where you really have hang on. Relax, you'll feel more subtle bites and bottom features. Second, don't allow too much slack to build up. Even the best rods fished with braid will have zero feel if the line is totally slack. This is where terms like "semi-slack" and "tight line" come into play. Third, keeping the rod angled up or to the side will transmit more "feel" than if you're pointed straight at your jig (remember, use the rod to move your bait). The fourth way that I was guilty of losing connection was that for years I failed to consistently watch my line. When fishing a semi-slack presentation, line watching is the difference between detecting subtle bites versus feeling the bottom. For this, braid has made life way easier.
I know this is a lot, but it should troubleshoot most feel issues. Also, as @adrenalizd mentioned, spooling with braid and fishing a FC or Copolymer leader is by far the most cost effective upgrade that will make a noticeable difference.