I'd suggest looking in the big box stores like Costco or a dedicated camera store like Cameraland at either a Nikon or Canon kit. They will have something like 28-55mm and 55-250 lenses. All are "auto" ready for us simple minded folks and have a ton of manual functions as well if situations get tough. They might just be pushing the top end of your budget, but one typically finds bigger is better in the long run.
Most can also do about 8 to 16 frames per second on "sport settings" depending on ones desired image resolution. Big megapixels counts obviously being a bit slower.
I'd suggest Googling up the tech sheets for several models, nailing it down to a few options, and then exploring prices.
Used or refurbished might be fine if they come with a company backed warranty. Given the dollars out, I like to have some time to put things through their paces and secure a fix if an issue pops up. If a new purchase, one might consider an extended warranty, again if it's company backed. Few if any of the extended warranties offered by second parties are worth the paper they're printed on.
With baseball and eventual desires to pull in fielding images one might want to step up to something like a 100-400 lens, but one could wait for a year or so to build up a little cash. The good, bad, and ugly facets of the hobby are that one can always add to the equipment bag if a family member really gets involved.
Some years ago, I created a monster by picking up a camera for Cookie. The next additions to her kit just might need a cosigner, and she burns more fuel chasing wildlife than I do in a full year of hunting/fishing.
Good luck and have fun.
Last edited by 1minute; 07/08/14.
