The First Day
As exciting as it is to bring a new dog home, the first day (first several days, really) should be as tranquil and disturbance-free as you can make them. Remember: your new foster pup has just finished an exhausting journey over hundreds of miles and several weeks of stressful, unpredictable changes (losing his original home, possibly spending who-knows-how-long as a stray, getting picked up by animal control or the shelter's front desk, spending days or weeks in the noisy confines of a shelter kennel, going elsewhere for a 10- to 14-day pre-transport quarantine, and then getting uprooted yet again to travel north). He doesn't know who you are, he doesn't know what this place is, and he doesn't know why he's here or what's going to happen next. The very first thing you need to do is establish a consistent routine in a way that lets your foster pup know he's landed somewhere safe.
Actually, the very first thing you need to do is get those ID tags on your foster mutt. Because he's been through so much upheaval, the odds are high that he'll run off in search of his old stomping grounds if given a chance. Don't give him that chance: keep his leash attached to a harness (it's more secure than a standard flat collar, which can slip off a thick-necked dog if he really pulls; my very first foster dog slipped her collar two minutes after getting out of the car, necessitating a super fun panicked chase through Philly alleyways at 10:30 p.m. I got her back -- running away with a handful of treats and enticing the dog to chase works wonders -- but whooo that was terrifying. So: harnesses!), and keep his ID tags on.
The second thing you'll probably want to do is let the dog outside for a potty. I'll cover this in more detail in the Housebreaking section, but especially in the beginning, it's important to take the dog out frequently (as in every hour on the hour, if not more often), let him sniff around for as long as he needs, and always be ready to reward heavily for pottying outside. It may take a while. Many dogs won't pee when they're nervous, and it's not unheard of for newly arrived fosters to hold it for 24 hours or more. You can't force your dog to pee; all you can do is give him lots of chances, lots of time, and lots of patience.
And then it's time to bring him in. Let him sniff around the rooms that you're initially allowing him to access (anywhere you don't mind messes too much); show him his crate and entice him to step inside by tossing a few treats toward the back. Introduce him to the house's dogs and cats when he seems ready. Show him where the water bowl is and feed him his first dinner. Offer soothing words and gentle stroking if he seems receptive; most mutts are eager for the reassurance, but some may not want to be crowded. Don't pat him on the top of the head, hug him around the chest and shoulders, or reach for his face right away; while many dogs don't mind those particular gestures of affection and some enjoy them, other dogs find them slightly to highly objectionable. You're still a stranger to this mutt, and until you've established a little more rapport, err on the safe side and don't give him cause to be annoyed or alarmed at your handling.
Let your dog's responses be your guide in those first few days. It's a good idea to take him on short, frequent walks (10 to 15 minutes or so) around your immediate neighborhood to orient him to the area and encourage pottying outside, but try to avoid areas with loud, intimidating stimuli -- construction sites, soccer fields with lots of children screaming and running around, yards with aggressive fence-charging dogs, and so forth.
You and your foster pup may get comfortable with one another within hours. Or it could take a couple of days for him to warm up. You probably won't have a severely shy or fearful dog your first time fostering (or ever, unless you ask for one), but if you do, it could take much longer (as in weeks to months) for the dog to relax around you. In most cases, however, you should be ready to move on to the work of training and socializing your foster furball within a few days of his arrival.
