I just moved to Jacksonville, Florida, in March and I spend a lot of time by myself. Here's what I do to take up the time other than chilling at home or on my computer....
1. Be a tourist...I like to travel around within an hour or two of where I live and see whats out there. I like driving up and down neighborhood streets and I also like taking pictures of cityscapes and such. St. Augustine is 25 minutes away and it has a lot of neat shopping there.
2. Go to the beach and get some exercise walking up and down the coastline. When I lived in Cincinnati, i walked along the riverfront downtown on the Kentucky side of the river cause there was no beach. Water is so relaxing and motivating.
3. Go to the gym. Usually they have cardio or yoga classes where you can be around other people. You'll feel better about yourself after you routinely get a nice workout. YMCAs have a lot of fun stuff too, if you can afford a membership.
4. Take dance lessons...most studios have a dance party Friday or Saturday night so you can practice with others that are learning to dance. I've been taking Latin dance lessons for three months and they always have a party on Friday nights which they expect me to be at. This option can be pricey though.
5. I spend an hour or two at Starbucks. The one near me is cool cause it has a bar that you can sit at and chat with the baristas. After going in there about 3-4 times a week, I became a regular and they all know me.
6. I like going to airport viewing areas to watch the planes take off and land. Good for about 30 minutes on a nice evening. Most airports have these and the planes go literally over your head.
7. Speaking of airplanes, most small airports that have flying schools will offer a one-hour ride with a flight instructor. You pay like $89 and get an idea how to fly a small 2-seater airplane. Usually the instructor will let you take it off and steer when you're in the air, even on your first flight. You get a neat perspective of your city too. Bring your camera. If you have the money, time and desire, take flying lessons and earn your private pilot license. Then you can take your friends up too.
8. I always like going to a playground by myself when there aren't others around and jump on a swing for a half-hour. Its a wonderful way to daydream and relax on a nice day....though the little kids may look at you weird.
9. I also like going to a park that has ponds and search for alligators. I love the thrill of seeing a big alligator in the wild...especially when he's interested in you.
10. If you have a river or large water body close by, I'm sure there are places where you can rent a canoe or kayak for a few hours. Its neat to kayak near the shore and enjoy the tranquil beauty and peace of God's creation.
11. Volunteer at the animal shelter on a Saturday and play with dogs an cats for a while. It feels good to see them enjoy getting out of their cages and having a good time with you. This is also a fun thing to do on a date.
12. Usually most cities have a Habitat for Humanity that meets on Saturday mornings. You volunteer your time to help build a home for a family in need. You don't have to know which end of a screwdriver to hold...they'll teach you. I learned how to do some electrical work, all the while the training was free and I was helping others.
13. If your city has a Children's Hospital, then chances are they have a Ronald McDonald House. Volunteer your time playing with the kids, cooking meals, or cleaning rooms for families who have children in the hospital. I've spent some rewarding evenings there, especially on Christmas Eve and other holidays when I couldn't be with my family.
14. When its football season, I'm always at a game or at a bar watching one. Although I don't drink, I love going to a crowded sportsbar on Sundays and having buffalo wings and nachos while watching as many as ten games at once on the bar's 55 TV sets. I used to go to Bengals games 3 times a year. Now its going to be Jaguars games. Better get a Fred Taylor jersey soon...preseason is right around the corner.
15. Sundays usually start with 9 AM Sunday School with the singles and then worship service at 10:30. Afterwards the singles group usually goes out to lunch. Then there is a Sunday evening service too and our pastor always give a new message from the one earlier in the morning. So that usually takes up a good chunk of Sunday....except during football season where the sportsbar is calling my name at 1 o'clock.
Hope some of those ideas spark some adventure for your weekends. Get out there and have fun.
Sat. nights at home in front of the TV are getting old. So thought i'd see what the rest of you do on the weekend. 
