Can primary care doctors test my bloody urine to make sure it isn't cancer, or do I need to see a specialist?

No. If you have blood in the urine, you need a complete workup: 1. Upper tract imaging -- most commonly a ct urogram. 2. Urine test for cytology (look for cancer cells) 3. Flexible cystoscopy to look at the bladder #1-2 can be ordered by pcp, but cystoscopy is only done by a urologist.
Needs evaluation. Blood in the urine can come from a variety of sources of which cancer is only one. Your primary care doc is a good start if this is your first symptom but do not delay in starting a work up.