It takes time for any budding musician to be able to work on their own. At the beginning the guidance of the teacher is so important for accomplishing anything. But there are a few simple steps you can take to begin becoming independent on your instrument.
Go to the instrument every day. If only for a few minutes, you can begin to set the habit at the beginning of learning and build playing into your daily routine.
Do your warm ups. These can be as simple as finger taps on a table (Place your hands on the table, curve your fingers so your palm is elevated as if holding a ball and your fingertips are on the table. Tap each finger 3 times, work to not allow your other fingers to lift up.) Your teacher will have assigned something simple for you to do every day to build the necessary muscles and finger independence for playing, so make sure you do it!
Study Flash Cards. Learn the terminology you need to know in your lessons- the more language you know the faster you will understand music!
Spend some time just playing around. Improvise, play something simple, even if you feel silly. Get comfortable being at the instrument and making sound- no matter if the sound is good or bad!
Turn off your inner critic. The best quote I have that applies to learning a new skill is "be brave enough to suck at something new." Beginners will make mistakes, they will not be great, they will suck. That is all part of the process, so stop judging yourself and allow this new skill to be bad for awhile- it's simply one stage in a long process of learning! The reward in the end is totally worth it.
Record you lessons. If you are struggling with remembering an exercise or how a song goes, or where to place your hands, a recording of your lesson will serve as a very helpful reminder!
The most important part of learning to play music is simply to set the daily habit of playing. Don't judge the quality of the playing, there is plenty of time for that. You simply have to start somewhere, and in a few years you can look back at where you started and how for you came!