No you do not need to change the gravel it's actually better to leave it because it grows beneficial bacteria. You also can use the same tank. Since you have two tanks you might want to start learning about doing a planted tank. You would still need to water changes once your plants are growing well it will become easier and you won't have to do 100% water changes. For a small tank you can use an CFL bulb and a desk lamp to help the plants grow. You will want to visit the planted tank section for advice on how to get started. You can keep your betta in one tank until the plants are well established. You don't have to but I'd suggest it because sometimes they don't take off right away.
Otherwise for a small tank what I've done is stirred up the gravel really well and dumped the water then I take declorinated water that I've left to sit overnight in gallon jug and pour a few inches shift the gravel back and forth (side to side for a rectangular tank and in circles for circular tank) to agitate any poo that may not have come out the first time and I dump. Then I do it again. I probably would do it 2 or 3 times. Then you can refill.
(Other posters have said adding a filter would help and that is true.) You do have to be sure to clean well.
You can also invest in a gravel siphon. I used this type:
It works for a rectangular tank but for a round tank especially a planted tank I end up just using the tubing. This is especially handy for doing 50% changes. Some people just use airline tube I don't find it's easy to start the suctiion to drain the tank with airlline tubing especially for a beginner. Others get the same size tubing that is on a siphon hose from the hardware store. That is a good investment if you decide to do a planted tank.
However, if you think you might eventually upgrade to a 5 gallon with gravel only--- the gravel siphon I suggested is better money spent because it's great for sucking up debris.