I might be safer than you think. I used to refuse to ride in a city because it seemed dangerous and many people I knew who rode in traffic had been in accidents with cars. But when I ended up on a bike trying it out, I found I could almost always (~95% of the time) avoid traffic, even without bike lanes.
Side streets and alleys are great - too slow for cars, but you're probably cycling at 10-15 mph, which is fine. Parks can be great - a shortcut with no cars and beautiful scenery, while the cars have to go the long way. Campuses, plazas, etc. etc. Even sidewalks when empty or nearly-empty can work to bypass traffic or go the 'wrong' way.[0]
[0] Second to riding, cyclists love to lecture others on how to ride. And a favorite outlet for their obsession is sidewalks. They don't discuss or consider the reasons or merits, they just have found an easy outlet for self-righteousness and repeat the same phrases over and over. Just remember that sidewalks are for pedestrians first; you are a visitor. Give them the right of way always, let them know when you're behind them (e.g., 'on your left!'), and give them plenty of space (trying walking when a bike buzzes past in either direction - from behind, you can't even hear it coming). Defer to their safety, real and perceived. It's easy. No problems at all IME, except self-righteous people over-excited at an opportunity to lecture someone.
Side streets and alleys are great - too slow for cars, but you're probably cycling at 10-15 mph, which is fine. Parks can be great - a shortcut with no cars and beautiful scenery, while the cars have to go the long way. Campuses, plazas, etc. etc. Even sidewalks when empty or nearly-empty can work to bypass traffic or go the 'wrong' way.[0]
[0] Second to riding, cyclists love to lecture others on how to ride. And a favorite outlet for their obsession is sidewalks. They don't discuss or consider the reasons or merits, they just have found an easy outlet for self-righteousness and repeat the same phrases over and over. Just remember that sidewalks are for pedestrians first; you are a visitor. Give them the right of way always, let them know when you're behind them (e.g., 'on your left!'), and give them plenty of space (trying walking when a bike buzzes past in either direction - from behind, you can't even hear it coming). Defer to their safety, real and perceived. It's easy. No problems at all IME, except self-righteous people over-excited at an opportunity to lecture someone.