I would suggest something like a dinner out at your favorite spot or maybe dinner and tickets to a show, concert or sporting event you will both enjoy.
You can check out what is happening in your area and purchase tickets online at Ticket
We wouldn't do cards and presents though and we would only do it yearly not monthly. got to hold hands the whole time.) Then I took her out to lunch, then to a movie, then to a jewelry store to buy her a bracelet she liked - then to dinner, then took her back to my place and we made love several times, then took her home and left her with some roses and a kiss.
The whole thing cost me about $450 USD But shit, I like to spoil em. I think we'll start slowing down anniversaries when we reach our first year, and then we'll celebrate it every year.
The first month anniversary gift should be romantic, thoughtful and not over the top.