Is there a way that I can buy more property around my house?

No, I don't think there's a way to get more land. It would be really helpful though! Maybe you can put the house you made in your house thing (I don't know what it's called! >_< ) and then put it on a bigger lot...

Actually you could go to "Edit town" and then just place a bigger lot. Then "move" your house to that bigger lot.
Erase the original NOW EMPTY lot (where your house was). Place a bigger one and move back your house to it's original location (or you could just leave it in another part of the city).
Yes there's a little work to do, but you can actually have a bigger lot in exactly the same location (so distances from store, school, work, etc. Will be the same)

Well I don't think you can make your lot bigger, but you CAN go onto town mode, click on your house, and when the little blue box thing shows up around the bottom, click save to file or something like that, and then go on the bottom and find your house, click on the house and drag it to a bigger lot, hope it helped!

When you start by a cheap house either then either use cheats or wait to get more money and go on computer and click move, move to the house near the beach for a bigger lot, it also comes with a pool and a big 2 story house. Hope I helped!

No, but you can buy a lot next to your house and just make it part of the house you have now, if there is space. Or move to a bigger lot