Hi Geoff: Congrats on your first house. You'll have a great time working on it, building into it your personality and tastes. Its very gratifying, although at times can fully test your will :-)
You don't mention if these bath fixtures are floor standing (as with a sink cabinet or set of shelves) or wall mounted. In either case though, your best bet by far is to secure things to studs....and if they are wall mounted (towell racks, soap holders or shelving), this is a must. Plastic drywall (not masonry!) anchors are ok for lightweight picture hanging...but thats about it.