Can you get tested for all std's and sti's by blood work and urine test?

Not quite: Urine testing (or vaginal swab) checks for gonorrhea and chlamydia, and blood tests are for syphilis, HIV, and sometimes herpes (HSV) and hepatitis B (depending on risks, symptoms, etc). Testing for HPV (with pap smear), trichomonas, and other vaginal infections requires pelvic exam. By the way, STD and STI are the same thing; there is no difference in meaning between the two abbreviations.