It's really hard to find new and entertaining things to do when you are on a budget. I find myself going to movies or coffee shops to get out of the house, which is not the most frugal thing to do.
So here a few things that I would recommend that you can do to entertain yourself cheaply.
1. Urban Exploration
Pick an area of your city or town that you have never been to and go and explore it on foot. If you are a budding photographer, take a camera. Look for nice trees, odd mailboxes, strangely named stores.
2. Join a Meetup Group
There are group for just about every topic in just about every area. Maybe you could join a book club or a philosophical discussion group, or take tours of peoples gardens. The possibilities are really endless.
3. Go to College Theatre
Most colleges put on one or two theatrical productions a year, and most of the time they are free or cheap. College productions may not have elaborate sets and costumes, but they tend to take more risks with production types, so you can often see some meaty material or a new interpretation of a classic. Also, colleges tend to draw smaller crowds, so the cast and crew will often greatly appreciate your presence.
4. Fly a Kite
No one seems to fly kites anymore. It's sad because kite flying can be done anywhere that you have an open space, and it is a lot of fun! If you want to invest in a more expensive stunt kite I would recommended finding friends to split the cost with you. If you are looking for something a little less expensive, though, you can try something like this well rated Rainbow Kite.
Something in your community is bound to need volunteers. Be it a community center, a park, a zoo, a theatre, a school, an art gallery. This is a great way to meet people and experience something different, and it's good karma too!