The best advice I can give is to make sure everything you choose can do double duty. "Work horse" furniture in small scale. A trunk for a coffee table or side table, so boys can keep toys inside. A sleeper sofa, to double as a guest bedroom. A larger ottoman can double as a coffee table as well as additional seating. Graduating in size baskets with lids, can be a decorative item as well as a place for storage, be it for linens, off season clothes etc. Small spaces require smart storage solutions to not look cluttered and to allow you to have the best free floor space available, especially with children and dogs. Look at storage options from the top down. There are some great under bed storage boxes (some on wheels) for clothes, toys, linens and shoes that may free up closet space for larger items.
Sofa's with rolled arms take up more floor space with less seating available than a sofa with straight arms. Armless chairs are visually unobtrusive in a small space, and can easily tuck into the smallest corners; you should be able to find a spot for them easily when this is no longer temporary. Because of their size, they take very little fabric to reupholster down the line. There is no use in buying too much that can't be incorporated somewhere down the line when you have more space.
This is a great time of year to buy small space solutions for childrens rooms (college dorm back to school sales) look for the loft bed options, they have desks and sometimes dressesrs underneath to maximize space. Captains beds have drawer storage underneath, or a twin with a trundle can free up floor space yet have a space for sleep overs or for 2 sharing 1 room. A tall chest of drawers is a much better option than a low dresser and mirror. Hang a mirror on the back of the door if needed. I hope this helps, and good luck.

I agree with trying to find furniture that serves multiple functions, also keep the color scheme simple.

Really good advice from shawnlagemann. We lived for many years in an 1100 sq. ft. house that we adored, though there were only three of us plus one medium sized dog. I wholly endorse the suggestions to go vertical with bedding and storage, go small on furniture size, and make sure everything possible does double duty (for example, we love antique footstools and make sure all of ours have lids that open for storage; our master bedroom had a platform bed with four huge drawers beneath; our son's bed had large storage boxes on rollers beneath it). These days there are some great murphy beds that can do double duty as desks or even dining tables when they are shut up. Even though we now live in a larger house, I find that the techniques I learned for maximizing space (as well as some of the double duty furniture) have proved very useful for any living situation we are in.

Best of luck with your downsizing move!

P.S. I suspect your mastiff will adjust beautifully, though his body and bed alone will take up some precious square footage. But I'd worry more if you had a crazy energetic breed that needed lots of space to run about and expend excess energy. Mastiffs are so nice and calm.

Dytecture posted while I was writing, and his advice about a simple color scheme raises an important point. The smaller the space, the more it will look cluttered by too many colors in the mix. So going light and simple in the color scheme will help keep the space feeling open and airy.