I need to know the best way to take up old vinyl and then what the steps are for staining the concrete in that area?

Depending on the adhesive used and the quality of the job, you may be able to simply pull the vinyl up or you may have to use a floor scraper, which is no more than a glorified and oversized razor blade :)

If you've scraped the floor well enough, with most adhesives, you can paint over the job with no further steps. Some adhesives may be more stubborn, however.

As for staining concrete, simply read the back of the package. It depends on the brand and type. Some are a rub on/rub off, like wood stain, and some are more like a paint.

Thanks,
Json