Congrats! I'm brand new to this too (haven't even finished my first project yet), but I agree with the poster who said to pick something that will KEEP you interested as your next project. Years ago my then-boyfriend's roommate tried to teach me how to knit by guiding me along making a dishrag. It looked a lot like your "before" pic, and it didn't hold my interest because I didn't really want a dishrag, so the motivation to improve just wasn't there. This time around, a different friend taught me and I got to work right away on a scarf. I have sooo much more motivation now because I actually WANT to finish it and wear it.
I feel like (at least for me) the key for the still-learning phase is to work on something that excites you, because it can be sooo frustrating in the beginning.
Then once the process itself becomes easier, taking on more difficult patterns or objects for gifts will be easier too. I don't know what my second project will be either. I'm only about 1/3 the way through my scarf. Maybe a hat? Or maybe I should do something that won't be in the round so I can get some much-needed practice with standard needles?