You can use computer in public library. I believe rules state that you must use computer to purchase real estate, even if there isn't much difference between using phone and computer (I haven't find any).
You'll see all properties and buildings available to purchase, and have separate list for real estate you already own. It is easy to navigate, and I'm shore you'll manage.
Once you purchase property, go in town view, click on icon of that property, check requirements for upgrade, and then chose in same menu to build/buy on that lot. Usually requirements are just to reach certain value of property, but sometimes you need to place few picnic tables, etc. It is all listed.
Good luck.