Hi auntcarrie4 and welcome!
I think your site idea is fabulous, especially since there aren't any in your area that you know of. Local advertising will definitely be the way to go and I do think SBI! can help you get started. You would use the brainstorming tools to determine if there are enough searches done for "your city + family" or whatever keywords you'd be targeting.
You can also start by using AdWords' tool to see how popular it is...
https://adwords.google.com/select/KeywordToolExternal
Now, in terms of the calendar, that will require some additional programming if you want other people to be able to post to it. Not sure SBI! is the best platform for that kind of functionality due to some software limitations at the moment.
So that leads me to think you may want to try a regular host if you're thinking of adding some kind of user interactivity with calendars, etc. I'm no programmer, so I'm not exactly sure what's the best way to build a searchable calendar like that, but maybe some more of our technically savvy folks can offer advice in that arena.