Newbie investor- Current tenants?

[Sam Webecke Halversen]

We are under contract for our first rental property (yay!), so now the real work begins. I would love to hear your feedback on a couple questions.

How do you introduce yourself to the current tenants? 

What ways have worked for you to raise rents on month to month tenants who are under market so the unit is left intact?

I'm thinking we will open a new bank account and run every thing through there to make book keeping easier- is that the best idea or is there something better?

A little about the property if it helps: Its a triplex built in 1940, 2/1, 1/1, and a studio apartment. 2 tenants are on month to month, but the other just signed a year lease in January. We don't mind him staying- although his rent is under market, he takes care of the unit and pays on time from what we can tell. Rents are currently 600 for the 2/1, 475 for the 1/1, and $400 for the studio. 

Market rents for the area are $700 -$800 for th 2/1, $500-$600 for the 1/1, and probably around $425-$450 for the studio (there aren't that many to compare). The 2/1 and studio would need a little work before we could probably charge the higher number for market rents.

Thank you all for your input and knowledge!