I am not a doctor. Instead, here is advice from the WHO.
Their first recommendation is to wash your hands. Wash long with lots of scrubbing for 20-30 seconds. Scrubbing your hands under running water may be more important than soap[1]. (Use soap anyway!)
If you are washing your hands frequently and maintaining social distancing to the extent you can, then you should pay attention to not touching your face. We all touch our faces unconsciously. It's easier to do something consciously (like wear goggles) than to not do something unconsciously (like repeatedly choosing to not touch your face).
I would bet many points on Metaculus that if goggles are in place over your eyes then you are less likely to touch your eyes with your hands.
By the way, here are the WHO guidelines for how to use a mask properly.
These details about hand-washing come from recent episodes of This Week in Virology, a podcast about virology by virologists. ↩︎
As far as I know, no literature is out designating infection to exposure in the eye. Regular mask usage would not protect you as much as proper sanitation would. I wouldn't recommend public transportation but obviously, different circumstances (financial and what not) predispose people to choose a certain mode of transport. I would recommend Uber or some other form of isolated travel for the duration of this coronavirus scare.
FYI, regular surgical masks are insufficient for protection against COVID-19. A respirator graded n95 or higher is required. Not sure what you mean by "surgical mask", but want to make sure they're not regular surgical masks.
Source: https://www.livescience.com/face-mask-new-coronavirus.html
[EDIT: You probably shouldn't read this comment, and instead read this post by Scott Alexander.]
FYI, regular surgical masks are insufficient for protection against COVID-19. A respirator graded n95 or higher is required.
Disclaimer: I'm not an expert.
[EDIT (2020-05-30): you really shouldn't use the following for updating your beliefs.]
After a quick look at some of the papers mentioned in Elizabeth's answers here I updated away from the belief that surgical masks are substantially less effective than N95 masks at preventing the wearer from getting infected with the novel coronavirus (it now seems to me likely plausible that surgical masks are not substantially less effective). But I can easily be wrong about that, and the evidence I've seen seems to me weak (the papers I've seen did not involve the novel coronavirus).
I'm in a high risk group and there's a higher likelihood of dying if I get infected then for the average person. I will reduce the amount of public transportation travel but being totally isolated won't be good for my health either.
I have a decent supply of surgical masks. How much percentage of the infection risk is do to having exposed eyes? Does it make sense to protect them with swimming googles?