Stuff to do in/around Trieste?

I have an opportunity to tag along on a business trip to Trieste, Italy with OH later this year. Reviewing Lonely Planet and TripAdvisor, it seems there are some things to do but not a ton. Sounds like getting around can be confusing. I'd be on my own most of the time, without a car, and my Italian is pretty much limited to ordering a pizza.:msnblush:

Is public transport decent? Reasonably inexpensive? Is it safe for a single gal to be wandering about like a lost tourist? Any options for zipping over the borders to Slovenia and Croatia to do something nice?

Thanks for any suggestions! :)