I have been needing a new bike and I need some help with what kind of bike I should purchase. I plan on riding on mainly paved side walks but when traffic is bad I need it to endure riding over curbs and in grass. I want it to handle long rides. I guess I want more of a commuter road bike. I don't really know the price range for a good bike, I just want the cheapest bike possible without it being cheap in quality.
Assumption for all that follows: You're looking for a brand new bike as opposed to used but new to you.
If you want the most bang for your buck, a mountain bike is probably the way to go. Mountain bikes are by far the largest portion of the bike production industry and mass production makes them cheap. Their upright geometry and wider tires make them great for getting over curbs, maneuvering around obstacles, riding off-road (or on crappy roads). You can get an introductory level mountain bike from a bike shop for about $350.
The next step up in price range will be a "commuter" or "multi-purpose" bike. (Both of these are foolish names since any bike fits those descriptions.) The Trek FX and Specialized Crossroads are both perfect examples of this class of bike, but there are plenty of others. These types of bikes will still have that upright mountain bike feel to them, but the larger radius and narrower profile of the tires will make them significantly faster. Also note that both of the bikes I mentioned do not have front shocks. From what you describe, I don't think you need one but similar bikes with shocks do exist. Bikes in this class start out at around $500.
For the type of riding you're describing I would recommend a cyclocross or a touring bike. They're so similar as to be almost indistinguishable; the most noticeable difference being that touring bikes have more places to mount water bottles and racks. Either of these will be more expensive, but will also be significantly faster than a mountain bike or commuter/multi-purpose bike on almost anything but true cross country trails. The tires will be the same as the ones on the multi-purpose bikes but cyclocross bikes come with knobbier tires, a characteristic which you may find desirable for riding through the grass. If you really want those extra places to mount racks, you can fit knobbies on touring bikes easily enough. You can find cyclocross bikes for $800 but $1000 is more the norm. Touring bikes are almost always $1000-$1500.
I personally wouldn't recommend a genuine road bike for what you're talking about. It won't handle grass or curbs well at all. Road bikes also tend to be at the more expensive end of the spectrum. You can get cheaper ones, but they sacrifice a lot of quality to get those prices down. Those sacrifices translate into a heavier and more sluggish bike, which is exactly the opposite of what you want in your road bike. Unless you get an older one on Craigslist or something. Older road bikes have larger tires and are more capable of what you're wanting to do with it. You can also find decent quality older bikes for pretty reasonable prices.
There are, of course, more exotic options. Folding bikes and cargo bikes and 29ers, oh my! Most of these are going to be pretty expensive and probably not what you're looking for.
There are also numerous options for everything I've mentioned above. Mountain bikes and commuter/multi-purpose bikes both have the options of shocks. Almost any type of bike has fixed-gear, single-speed, and geared options. You can put slick tires on almost anything. You can put knobbies on pretty much anything but a road bike. Some of these options will bring the price down, most of them will raise it. And a lot of those type of decisions will come down to experience and personal preference.
A road bike is going to be your best bet. I have been using my $300 mountain bike, but they need to be repaired so frequently I have switched to a road bike. As for cheap, Craigslist has always been a preference of mine.