Going to close on a triplex next week, but am going to have a conversation with the current owner.  Is there anything I should ask her? Should I start new rental agreements (they are all month to month)? Etc.  Any input would be greatly appreciated!

I'd have them sign a new lease with you.  Ideally, set the term to end near peak moving season (May-August?).  With a healthy advanced notice, let them know you're raising the rent to market rate.  Hopefully they move so you can have your own, personally screened tenants.  If they don't move, at least you get more rent.

That was my initial thought, but after looking at the tenants, I don't know if that is in my best interest.  These tenants have a long track record of paying on time.  The newest one has been living there for three years, the next at four years, and the last at 14 years.  Their rent is incredibly low, I could easily increase it from $480 to $550 and still be under the market rate.  However, if I do that, I could lose these tenants and this is my first time as a landlord.  At their current rates, my profit projects will be low ($170/month), but when and how much should I increase their rent?

If you raise the rents where are thy going to move to, don't think they will find anywhere cheaper do you? I always preface the rent increase with a letter saying I'm the new owner and unlike the old owner I have a mortgage payment he didn't have and will be having higher expenses then he had so I am going to have to increase rents by 10% or to Whatever starting the first of whatever. If you are planning to stay you will need to sign a new month to month lease agreement with me before whatever. They will check rents elsewhere then come in and sign a new lease. Don't go over board on the first increase but enough to start a move to market. I would want to keep them if they are good so I would work with them. I used to move to almost market in one swift move but found I liked this better overall. keeping it month to month allowed me to change my mind.

I really like that thought.  I will most likely follow your steps and act on it January 1st.

What do I do about the my tenant's initial check in (where they note any damages in the apartment before they moved in)?  I'm not sure if the current owner has them on file.  If not, are these people exempt from any/all damages? 

I would just bump the rent $10-25 each year until you reach market rates.  If you have good tenants you don't want to drive them all out at once. 

Thanks Bob.  

Should I do any improvements first to justify some of the increase or just do it because I can?