Existing answers cover most of your question well:
Your mother's insurance rates definitely can rise if you move into the household and have a driver's license of your own
It wouldn't be difficult for the insurer to find out that you have a license, though they may or may not know that you live with your mother (particularly if you aren't a minor)
But they haven't covered the remaining question about you having your own car insurance.
It probably won't help your insurance situation much to have your own car and coverage. That you might drive your mother's car is what would cause her rates to rise in this situation, representing some nonzero additional risk to the insurer.
Having your own insurance mitigates that somewhat, but doesn't eliminate that extra risk to your mother's insurer (coordination of benefits for auto insurance gets messy quickly, and it's easily possible for your mother's insurance to have to pay out even if your own coverage does so also). Talking with the insurer is the only way to know for certain.
An option that may exist for you and your mother is for her to speak with her insurer and get a policy which specifically excludes you from driving her car. A consequence of this is that you would effectively never drive that car, as you would be liable for any incidents that come up (either personally, if uninsured yourself, or through your own auto insurance if you have it). The terms of such an insurance policy may also state that you cannot drive the car under any circumstances.
Availability of such a policy will vary by insurer and state, but it may meet your and your mother's needs if it's an option for you.