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I bought a digital tuner btw, is it the ukulele or should I get a better one? How can I improve the sound?

IMO, at this point, I wouldn't concern yourself with sound as it's just fine with headphones (most people listen this way)...Just practice and learn the songs well and your performances will get better.
Getting better sound requires better mics,preamps, and a DAW--plus a good ear and audio knowledge. That can come, but, IMO, you're better focused on capturing a good performance and you'll have a better chance of doing that the more you practice each song. Hey, it will keep the diabolic busy:D

Ok, I've been practicing another song lately. This advice surely motivates me! Thanks!