Ok, I'm not sure how much guidance you want or what your experience level is etc so do forgive me if I patronise you - I've been biving around the place for years and with the dog for the last few - here we go;
First of all a disclaimer - Dartmoor is a high moor close to the sea and as such has rapidly changing weather, i
always take a windproof/lightweight shell. When the fog/mist rolls in the lack of contours can make navigating a real challenge (IMO the area north of redlake is one of the toughest micronav sites in the uk) - that said, you're never more than 10K from a road so if the worst happens self-rescue is pretty easy. Please make sure you have a map, compass, emergency food (and gps if you have one) with you.
Sorry about that - had a bit of a reminder about safety yesterday - I arrived back at Heytor just as this was kicking off;
http://www.bbc.co.uk/news/uk-england-devon-23286952
For a dog assisted bivi I usually take a few extra things;
extra sleep mat
extra first-aid
Dog boots
packable bowl
Stuffed meat-filled bone as a treat (and entertainment, perrin has a short attention span and it's great to have distraction free pitching)
if it's cold then i take a cut down old synthetic sleeping bag for him.
Larger tarp so that there's room for us both to shelter - the dog will try and join you in the shelter anyway, you might as well design it in.
If it's bad weather I take my wild country one man tent (sadly discontinued) as it has a massive porch, big enough for bags/cooking and dog (black lab) without being too heavy.
I've tried to suggest a few routes rather than dictate/plan it all for you but I'm happy to share .gpx if you'd prefer and i'm currently trying to build a list of good bivi grid ref.
If you don't have a map to hand then;
http://maps.the-hug.net/ is brilliant for online and for a physical map the harveys mountain map is printed on plastic instead of laminated (so it lasts more than a few days in the rain) and the 1:40000 scale is perfect for bike or longer trips on foot (IMO).
Probably the easiest bike-camp-bike that's 100% offroad is to take the old railway line from shipley bridge (680 629 - shorter) or from ivybridge (longer) up to redlake (646 688) and cruise back again. There's not huge amounts of water on the way there and back for the dog so take some extra and a bowl. the terrain is also quite hard - Perrin often wears his ruffwear boots for that one. (incidentally he usually has a ruffwear pack on too so he can carry his own water and food)
The princetown loop
http://db.tt/rpLb5w5P has great potential for swimming at the quarry and bivi anywhere in the southeast corner of the loop. I've not done it with bivi though I have bivi'd in spots on the route
The other good one as a first time is to do a loop out from okehampton camp to the cranmere pool letterbox with a camp somewhere near hangingstone hill (617 860) - as you can see from the map there's plenty of water about and the military "roads" make for good cycling. I like to include a trip up yes tor when possible.
If you don't mind the odd road and road crossing - (I usually take perrin on the odd one or two) then the choices are endless;
How about something like this:
Start in widecombe, climb up over 707773 and down past blackington manner, up through sousson's down to the Warren house Inn for an evening meal and a few jars (673 780). Stumble out and you're close to the edge of the camping limit so head up the path to bivi somewhere like 660818. The next day head back to pub for breakfast then across to grimspound (702808) and over to jays grave (732799) there are a few streams along the way but I'd pack some water anyway just in case. From Jay's grave head to the hound of the basketmeals for tea and cake (740792) before sneaking quietly past 733789 or otherwise finding your way home to Widecombe.
Incidentally I've always fancied a bivi in phillpots cave but have never got round to it - that might have to be on foot though.
Hope that's enough to get you started. Please feel free to ask if there's anything more/else etc. I don't often post but I'm lurking around.