by "they are kind of worn" I assume you mean the brake pads are worn. brake calipers (the metal part) are much less likely to be worn out.
You're unlikely to get any improvement out of "upgrading" to Shimano brakes, beyond fresh pads.
So instead, you should just buy new pads for your Tektro brakes. Tektro has a good reputation for their brakes, but not so much for their pads (at least the low-end pads), I'd recommend Shimano or Kool-stop pads.
In general, Tektro and Shimano are very different animals.
Shimano makes a full range of bicycle parts, Tektro makes brakes and brake levers. They both make parts of varying expensiveness and quality.