Is grocery store garlic as good as  it gets or are there places to get higher quality garlic that will elevate ones cooking?  Would better quality garlic last longer? I don’t cook much and often find the center of the clove  has turned green when I go to use it. 
Garlic lasts a long time, how long depends partly on variety. Hardneck varieties have fewer, bigger cloves and generally do not last as long, softnecks have more and smaller cloves and last a bit longer, but they both last a very long time. I grow garlic and I'm usually still using my last year's batch when my next is ready, so you should be able to keep yours for weeks at least. If it is sprouting most likely you are storing it in the refrigerator, this is a common mistake as most people do not know that garlic germinates in cold weather. Store it at room temperature instead.    I've lived many places in the world and there's no single answer to the quality aspect as it varies from place to place. In most of the US and UK the garlic you get at the supermarket is pretty tame, bred for quantity over quality, whereas in Italy and France the garlic you get at the supermarket is far stronger - one clove of Italian garlic is equal to a whole bulb of typical US supermarket garlic. Organic varieties in supermarkets tend to be much better, you can also order online if there are no good local farmers markets or other sources nearby.   If you like good garlic and you have the space you could grow some yourself, that way you can pick a strong variety. It's actually very easy, tolerant of a variety of soils, you just need to make sure its well fertilized and not to let it get dry. Of course, depending on where you are in the world it may be too late to plant out as it needs cold weather to germinate.  