I use the bathroom more than 10 times daily lately. Would it be good to see a doctor?

Yes. Typically anything greater than 8 voids per day is consider high. It may be caused by a number of problems or simply be a function of drinking large amounts of fluid.
Yes. Urinary frequency could be secondary to a bladder infection, specially if burning with urination is also present. Let your doctor do a urinalysis and decide whether you need antibiotics or need further workup.