The first one is not grammatically accurate but it is widely used and accepted, especially in American English. You may also say "You should go to the doctor/doctor's", or "You should go and see a doctor".

source: https://dictionary.cambridge.org/grammar/british-grammar/common-verbs/go [text] Go and, come and … In speaking, we often use and after go and come before the base form of verbs like ask, buy, check, collect, do, find, get: Why don’t you go and ask her? I need to go and buy myself some running shoes. Can you come and do the washing-up? She should go and find another job. Spoken English: In informal American English, speakers often leave out and: Would you like to come have lunch with us? I was expecting him to go get the keys.

In the United States we would usually say "You should go see a doctor." "You should go to see a doctor," "You should go and see a doctor," and "You should see a doctor" are also correct and are fine to use. In formal English, you might say "You should consult a physician" or "You should seek a medical opinion."