I have been clicker training with Aspen and he understands "sit", but doesn't really understand "lay down". Should I be using a different term for laying down, such as just "lay" or "down"? Also, what other basic commands should/could I teach him? He is only 10 weeks so I just want to practice easy commands.
Ask a hundred puppy owners, get a hundred different answers. I think it's more important at this age to be teaching thing like being handled, having teeth, ears and eyes checked, filing nails etc. General handling for vet checks. Impulse control for not pushing through doors. Waiting until being told before taking food. Not mugging your hand for treats, but to wait until offered. Settling down and waiting their turn. As well as toilet training and bite inhibition. I think that's loads to be getting on with, along with play sessions.
I was always taught to use one word commands. So just DOWN. I do think it's a good idea to use words in vogue with your own training community and dog owners in general. That way when the Vet or boarding kennel or friend or person who finds your dog ( it happens) they will be able to communicate and keep him safe and less stressed.
Down took us ages to teach. She didn't really do it until around 4 months (and still isn't a big fan). I used a hand signal at first and then introduced "down". We started with sit, wait, and and a cue to go pee ("do your business") at this age.
To go back to an original point, when you say your puppy doesn't understand your verbal cue (whatever it may be), does he understand a visual cue? You should only be introducing a verbal cue when he's very consistently doing the behaviour when you "ask" in another way - which is usually an approximation of the original lure movement. So, when teaching a down to my puppy, I first pop her in a sit, then move a piece of food from her nose to the floor. She got this pretty quickly - my other two took forever. Maybe because I waited until she was older, who knows. Anyway, once she understood what I was after, I stopped using a piece of food, but faked it with the same motion, then giving the reward from the other hand. This motion is my thumb and first two fingers together, but otherwise an open palm, facing down and moving down. At this point, still sitting on the floor or crouching down. That's where I am now (I just started teaching it halfheartedly a couple of days ago). Once you have this, and he is reliably performing the behaviour when you make the motion, so that you would place a bet of £100 on him doing it, then and only then should you introduce the verbal cue. You do this by saying your word (I use "floor", because "down" is used for so many other things - "don't jump on me", "get off the sofa" etc; other people use "drop", but I use that to mean "drop something you're holding in your mouth onto the floor".... it doesn't matter what your cue is, just make it something clear and distinct), and, once you've said your word, give your hand signal immediately. Repeat a few times. The formula is the same whatever you're teaching; new cue -> old cue -> behaviour. So... say "floor", give the hand signal, wait for the behaviour. If you don't get the behaviour, you lose £100 After a few goes of new cue ("floor") -> old cue (hand signal) -> behaviour, you can add a pause between the new and old cues. Wait a few seconds to see if he has made the connection and anticipates the behaviour. If he does, throw him a party. If not, give the old cue and go back to repeating new cue -> old cue a few more times before trying again. In the end, he'll work it out. For me, I like to have a full visual cue I can use at a distance. So, I don't actually introduce the verbal cue until much later. For me, the next step is to start opening my hand so I'm losing the pretence of the lure. Then, I'll start to stand up until I can simply open my hand, facing downwards and move it in a downwards direction at my waist to get the behaviour. Once I'm fully standing and can just make this single movement, that's when I'll introduce the verbal cue, so I can choose whether to use verbal or visual. Since dogs tend to be more visual learners, I find they often respond better to visual cues than verbal ones when things get a bit tougher. But, most people are happy just with the verbal cue, which is why I would suggest, if this is you, that you can introduce the verbal cue without the faff of "finishing off" the visual cue.
Getting trained your puppies are really a challenging task. I like to train my dogs through youtube videos.
Lots of good advice from @snowbunny. Probably one word 'down' is best, but make sure you use a different word, such as 'off', when you want him to stop jumping up or get off the sofa. Some people seem to use the same word, which is confusing - and potentially just encourages lying on the sofa!
'Down' took us ages and lots of practice, so much practice that even now, 5 years later if he's confused and not sure what you're asking he'll go into a down. Important safety commands, Recall, 'leave' and taking treats gently would be what I'd focus on first. Hand signals have helped us a lot, our dog responds so much more to the signal than the word.
We say 'down' We lure at first with food, then keep giving them food in the down food to get them to stay down. Keep yourself and your hand low and close to the pup. As they sit back up we say 'upsit' so they learn that command at the same time as the down. The lure is the same as the hand signal so, when they have got the idea, we leave out the food and praise+treat after the action. We then start to add the word. When they've 'got it' we practice with hand signals and word separately so that they can do either. It does take a long time, but by five/six months old - when they do the bronze KC test - mine have all had a reliable 'down'.