Hello, my name is***** and I have over 20 years of experience as a veterinarian. I'm sorry to hear about your new fellow's diarrhea and one episode of vomiting. I am encouraged to hear that he is bright and active otherwise, those are positive signs.
Diarrhea can be due to stress, a change in diet, eating things that they should not (toy pieces, something in the yard), parasites, a viral or bacterial infection, pancreatic insufficiency, inflammatory bowel disease or even a food allergy. But rarely in a young dog it can also be related to metabolic organ disease (such kidney or liver disease) due to organ wastes irritating the gut or a mass in the gastrointestinal tract.
In his case because we know he is new to you, so he is stressed, and may be eating something different than he is used to it is probably related to those things but we do want to keep a close eye on him and make sure he comes along as he should with treatment.
A raw, irritated intestine can take a few days to heal. If not he should be checked by his regular veterinarian if he's not bouncing back quickly.
Even if he improves it is well worth having his veterinarian check at least a couple fresh stool samples too as parasite eggs/cysts are very common in young dogs and could be part of his trouble. They are shed intermittently so may be picked up on the second sample or third test if the first is negative.
If he still feels well otherwise now though we can try some things at home.
Since he vomited he likely has some stomach upset and reflux that can go along with loose stools.
To try and settle his stomach today you can give either:
1) Pepcid-ac (famotidine) at a dose of one 10mg tablet per 20 to 40 pounds of body weight every 12 hours.
2) Prilosec (omeprazole) at a dose of one 20mg tablet per 40 to 80 pounds of body weight every 24 hours.
These will reduce stomach acid and should help if this is related to simple nausea and gastrointestinal irritation.
I would not feed him any food for 24 hours after the acid reducers are started. This should help stop gut spasms and restore normal gut motility. Small amounts of water or ice cubes given frequently are fine as he needs fluids after all that he has lost with diarrhea. You can give him pedialyte to replace electrolytes too but Gatorade is much too high in sugar which can make his intestinal irritation worse.
Today even with the fast you can start Kao-pectate at 1ml per pound or 1 tablespoon per 15 pounds of body weight every 6 to 8 hours. This is quite safe and will coat his irritated gastrointestinal tract as well as absorbing bacterial toxins. You can use it for several days until his stools look normal. You can find kao-pectate at the drug store.
If he has a tense painful abdomen, continues to have diarrhea with no improvement after being on kao pectate for 24 to 48 hours, becomes very lethargic, begins vomiting consistently, or runs a fever greater than 103.5F then he really must be seen by a veterinarian as soon as possible. Make sure to take a fresh stool sample with you when you go.
After his food fast start a bland diet of 1/3 boiled, lean hamburger (or boiled, white, skinless chicken), all fats and juices drained off the meat, mixed with 2/3 boiled, plain white rice or pasta. Feed several small meals a day. You might wish to add 1 to 2 tablespoon(s) of canned pumpkin to each meal if you are seeing mucous in his stool as fiber helps soothe an irritated colon.
Diarrhea can also lead to bacterial overgrowth in the gut. Probiotics such as Fortiflora, Proviable, or Benebac can help replace appropriate bacteria. You can add one of those to the bland diet mix.
Once he feels better (no diarrhea for 48 hours) start mixing in his regular dog food very slowly, less bland more regular with each day. It should take a week or so to convert him back.
If your fellow begins vomiting consistently and feeling poorly, has a tense painful abdomen, or runs a fever (rectal temperature greater than 103.5F), then it would be best that he see a veterinarian as anything you give him orally will just come back up worsening his dehydration.
Please let me know if you have any further questions.