My wife and I have had some dogs that I thought were pretty smart, but our two-year-old Mastiff mix is super-smart, and seems to like learning new things. He knows “sit,” “lay,” “wait” (basically “stay”), “down” (to get off a chair or sofa), “drop” (to drop whatever’s in his mouth), “leave it” (to not put whatever’s interesting him in his mouth), “shake” (hand/paw), and when I tell him to find our cat (by name) or get his ball (“Get your ball”), he’ll find them. What else should he learn? I want to keep his mind active, and I want him to learn (mostly practical) new skills.
My step father had a dog that would get him a beer from the refrigerator. He could never get the dog to shut the frig door.
“Come” would be a good one, especially in an off-leash situation, or just any time you want him near you. Seems like that would be among the commands you already have rolled-out, no? How about “Walk?” or “Walkie!!”?
For our dog we also use “Breakfast?” and “Dinner?” to let her know it’s time for her to eat. We are a bit sloppy and sometimes use them interchangeably, but she knows both of them mean meal-time.
Teach him to pick up dirty laundry and put it in the hamper. Seriously! You’ll have to break it into steps and then chain them together. Any dog who will pick up what you point to, carry it as long as you say ‘carry’ and drop something where indicated can be taught many useful things. My Aussie Bonnie used to carry small packages from the store, and exchange them for a treat when we got to the car, and another one of my Aussies would fetch kindling for starting fires. I trained my childhood mutt to go to the end of the driveway and bring back the newspaper.
Another fun game is “find it!” Start with hiding treats or toys while your dog waits and then say “find it!” Eventually you can hide the treat in another part of the house. My Bonnie would generalize that command. She’d just start looking until she found something that seemed like what I might be looking for. On several occasions she found things I had dropped while hiking, like my hat once, and another time my phone.
Recall games are great and can safe the life of your dog some day. The idea is to find ways to get your dog to come to you faster and faster, from farther and farther away.
When waiting in line at Petco some years ago, I met a woman who had taught her dog the command “back up.” When she said it, her dog would scooch backwards a couple of feet. At the time, this seemed like a silly stunt, but now I think that’s one of the most useful dog commands I’ve ever seen. Sometimes you need a dog to give you a little more space. I don’t presently own a dog, but next time I do, he/she is going to learn “back up.”
FYI – it’s bad form here on the SDMB to post about a dog (or cat, I suppose) without contributing the requisite picture to the Pet Pictures Thread. 
One of my (now gone) dogs liked to learn stuff. We had “load up” for getting in the car, “hop up” to get her to jump up on what I was pointing to, like a big rock, low wall, the couch, etc. “Find it” as described above, “crawl” (starting from down and getting her to crawl forward), “back up,” “bed,” “high five,” “give me ten” and “Shamu” (jumping up for a treat).
If your dog will find the cat, I’d try having him find you and your wife by name as well. I’ve never gotten a dog to figure that one out.
I’m so jealous. I grew up with a super-smart poodle, and was expecting something similar when we finally got a dog.
But we could never teach him even the simplest trick. The problem was he was too cynical. I’d put out my hand: “Shake!”; and he’d sniff at my hand… nope, no treat in that hand, don’t need it. “Roll over!” “You first.”
My wife spent an entire summer trying to get him to chase a ball and bring it back. She’d throw the ball, and he’d practically shrug: “Huh, guess you didn’t want that ball, so why would I?” Over and over again for weeks…
She even wrote on our huge blackboard “GET TUCKER TO FETCH!”
I wrote under it “Stop trying to make Fetch happen.”
Property lines?
Teach the dog to stay on the property even if off leash. This can protect the dog. Not sure how to do it. If you have a full fence. Maybe try and teach it to not go out a gate without you.
How is the dog when walking with you? Good leash training is great. Stays at your side. Matching pace well. Not straying at will. Doing all that when off leash as well. Until allowed to go free in the right areas.
Consider what training will protect your dog from harm.
We actually trained our beagle with “back back back.” If she knew a treat was in the offing, she back up 20 or 30 feet.
We taught our 2 Chessies ‘get it!’ , ‘left’, ‘right’ , ’ go on’, and ‘good dog’ when they had found the right object . Both dogs would fetch all kinds of things that way. And boy were they pleased with themselves!
I’d start with beginning-level Algebra, or if that’s too advanced too soon, work on basic arithmetic. Once fractions and long division are mastered, then move on to Algebra.
I don’t think I’d go the academic route. It’ll be years before the dog is doing any cutting-edge research and is able to publish. I’d aim for something more practical; get the dog to do your taxes.
Oh, another one that’s kind of handy is cueing them to go pee. Doesn’t matter most of the time, but occasionally you need them to not spend a lot of time on it. For discretion and propriety, we use the phrase “go sniff.”
I agree. They’ll have a much more profitable career if they learn underwater welding.
Since he already knowns the ‘lie down’ routine, a nice and fairly useful progression to that is teaching him to lay flat on his side. I know many dogs do lie flat anyway when told to lie down, but having it on command means that in a less relaxed situation you can get them to do it a bit more easily.
It’s useful at the vet’s, and for trimming nails or examining the belly for fleas or injury. It’s helpful for ear-cleaning, and just as a general ‘ok, time to chill now …’ thing.
To initiate it, when the dog is lying down normally (both elbows, chest, and one hip flat on the floor, hind feet out to the side) take a treat and lure the dog’s nose back over his top shoulder. The head twist should make him lean back a bit. Get him started being comfortable with that motion first, don’t expect instant flat-ness (though you might get it!) Once he understands the general game of the hand/treat motion, then you can label it with a word - I use Side.
Someone above mentioned “back” or “back up”. I love that one, and use it all the time.
I have two border collie mixes. Very smart and very active dogs.
Heal is always a good one. Jackson heals on my right side, Lani heals on my left side.
Place is also a good one. Basically, it’s like wait, but in a certain place. Say a dog bed for instance. It’s nice for when you have company or are otherwise engaged.
Our German Shepard Kali eventually had NoseWork classes, but we started out with “find”. I would hold my gf’s hoodie while she hid in the house. I’d say, “find (first name)”, hold the hoodie up to her nose, and off she’d go.
The first few times my gf would initiate the search by calling Kali, but she quickly got the concept. Eventually my gf would walk off into the woods. I’d give her a 10 minute head start, then start Kali searching.
With NoseWork we used a scent (balsam?) applied to the cotton tip of a q-tip that was place in a perforated metal holder. We’d hide that and Kali would find it.
The best thing we taught our dog is “U Turn.” When he hears it, he knows to turn around and run back to us at full speed.
Also, we taught him the name for various toys and task him with bringing back specific ones. So we throw three tows out and tell him to bring back piggie, then gator, etc. it’s good because I find games where he has to think tire him out more quickly than just running around.
Another is warmer/colder, red hot for bringing back specific things. If I throw a ball and he didn’t see where it went, we play this game until he finds the ball. I’ve also used it to get him to bring me one sock, not useful…yet.
Shaping is an amazing tool that can teach a dog to do something you’d never think was teachable. In the spring we are enrolling Kizzy in a shaping class.
Another consideration is Canine Good Citizen Certification. It isn’t easy, but we did this with Kizzy and all the behaviors are useful in real world situations. We recently had Kizzy in a brewery and a woman in a wheelchair asked to pet her. She has specifically been trained to sit calmly to the side of a wheelchair, and the woman was impressed.
Once a dog has their CGCC, demonstrating just 5 tricks gets the dog AKC Novice Trick Certification.