Pinot Grigio . . . sigh. It's a much maligned and misunderstood wine. Unfortunately, most PGs that are widely available in grocery and drug stores are undrinkable, and have become a bland substitute for white zinfandel for most wine drinkers, or an entre to white wines, along with overly oaked, buttery cheap California chardonnay.
First, there does exist really nice, crisp Pinot Grigios from Italy's Alto Adige or even Friuli-Venezia Giulia region; go to a wine shop and they will recommend some. (I don't know what you mean by "biteyness," but I would consider the crisp acidity of a good quality pinot grigio to be a little "bitey." A chardonnay is not "bitey" to me.) Also look for Pinot Gris (as Davooda points out), especially from Alsace, it's same varietal, but comes from different regions. Second, there are many, many different white wines out there besides Chardonnay and Pinot Grigio. I encourage you to take advantage of the many free wine tastings at Whole Foods, Binny's, etc., to introduce you with low risk to those wines. One white that I find many occasional wine drinkers to enjoy -- not a PG or Chardonnay -- is
Naia Verdejo from Spain. It is under $15/bottle, sometimes under $10, and widely available.