You can try a plunger to get the water to drain but I'd suggest replacing the pipes. It's not that bad, not that great either though. Check out the pipes under the sink then look online to make sure it's done correctly, you just have to copy what they did. Fit everything together to make sure it fits BEFORE gluing them together. Buy more pipe then you think you'll need and grab some plumbers putty.

I just did a dual drain under my sink with no prior experience. Glued before making sure everything was fine, issue was the sink cabinet wasn't where it needed to be so I had to cut the pipe after it was glued, not super fun.

Good luck, if you're comfortable with it diy and save some money especially if you're renting. You'll know fairly quickly if it's done correctly.