People often consider magnification when choosing a bino, but seldom do they think about field of view (FOV). FOV is the width of the scene you see when looking through a binocular. The higher the magnification, the smaller the FOV. FOV doesn't matter much if you are looking at static things, but it matters a lot when viewing moving things, especially things that are moving quickly such as a race car, or a football player. For stadium/sports use, I use 6x, 6.5x and 7x, in 32mm or 36mm. These lower magnifications have the advantage of being bright for their size, having wide FOV's, and also having great depth of field, which means that you don't have to change focus as much when things move towards or away from you. With higher magnifications, you will have a narrower FOV which makes it harder to track moving objects, and less depth of field which forces you to continually change focus as things move.
If you want to keep the budget under $100, consider the 6x32 Leupold Yosemite. These are nicely sharp and bright, and are light enough that they can hang from your neck for hours without any discomfort. For a bit over $100, consider the Vortex Raptor 6.5x32. Same good attributes of the Yosemite, but a bit more magnification. This is the one I would get if I were on a budget.
Another excellent choice is the Atlas Optics Entrepid 7x36 that is found at the Eagle Optics website. These are Zen-Ray clones sold by EO. The Z- R version run about $300, but the EO versions are on sale right now for $149. These are exceptionally sharp in the central area, have very wide FOV's, and are also very bright. They outperform binos that cost more than twice as much. They weigh less than the great majority of 42mm binos. Besides making a great low-end sports bino, they would also make a great GP bino too. At the moment, they are one of the best buys in binos under $500.
If you feel that you need an 8x bino, take a look at the Nikon Monarch 7 8x32, which has a very wide FOV for a 32mm bino, and weighs under 18oz. If you need to stay under $200, take a look at the Sightron SII Blue Sky 8x32. If you can swing it, take a look at the Zeiss Conquest HD 8x32, which is one of the best binos of any size under a grand. It's resolution and brightness and FOV are to near alpha levels.
Lastly, if you don't mind trolling eBay for a while, see if you can find a used Meopta Meopro 6.5x32. When they were still being sold, they went for around $300. These have an excellent build quality and view. In fact, these would be the EOT suggestion if they were still being sold. These also have about 20mm of eye-relief, so can be used by most glasses-wearers.
I'm not going to discuss alphas, but if your budget extends to $1500 or more, let me know.