Everyone's taken these courses that works corporate. It's almost common sense.
Managers also are trained very well on these and aren't idiots. We know how to CYA. Majority of people responding to OP in this thread are giving him unrealistic advice recommending legal/HR correspondence this early, which I highly doubt will yield results. The % of those who are actually able to prove negligence and hostile work environment conditions is extremely low compared to the amount of reports. OP is also very new to the job. The manager, HR, and company will be doing what's best for the company and not the OP. OP also never mentioned anything about those racial comments being in writing, which i highly doubt would have happened (you'd need to be a whole other level of idiot to send racial comments in email) which will further complicate things for the OP if there isn't written proof.
If you're going to go to bat for these things, go to bat to hit a home run, and not a foul ball or strike out. Full documentation of multiple issues of harassment, racial remarks, e-mails to manager making them aware of the issue, no typo's or reasons to justify complaints from other co-workers regarding work quality, and then, only then, bring it to HR. This is how to pidgeon hole HR, management, and company into resolving this issue. Doing this however just puts a target on your back for causing problems and making people tiptoe around you if they know you are the one who purposely got someone removed who's been there for a while that people got along with (assuming other's liked co-worker). I would only recommend this if OP can't resolve things personally between the co-worker.
I had a underperforming direct report who reported me 11 times after their first 6 months on the job. I faced all reports with HR and they deemed all of them pretty much non actionable or irrelevant reports. The direct report even complained to my boss, who knew how much of a problem this person was to me and decided to shift their work to basically nothing. That person was laid off about 8 months later because they were doing work that really wasn't necessary anymore (she wasn't doing it anyway) and we didn't have budget allocated to that position anymore.
Again, my advice for OP from someone who has dealt with these issues as a manager and individual is earn your stripes and or resolve this directly with your co-worker. Hear his point of view if he has one for being an *******, address those issues and move on. If he's just an *******, stand up to them and see if they back down. Then finally, if all options are exhausted, continue to compile evidence (should compile this entire time as back-up) and once enough is gathered, bring it to HR.