Hi and welcome!
I adopted my Chi about 4 months ago and she is a great little dog! Mine gets along with the cats too - I think it's because they're the same size - LOL!
I use a slip lead on her sometimes when we go out in public or on walks but it's more because it's the perfect length and not for the correcting part. My goal for my dogs is to teach them to control themselves because the more they control themselves the more freedom they have. I hardly ever put a leash or a collar on them anymore. If I do it's usually because we "have" to (leash laws, you know).
I have taught her to do the "down" and I used treats and my other dog to teach it - lol!
I do short training sessions numerous times thoughout the day - my max time is 5 minutes, but it's usually less. I always want our training sessions to be fun and to leave the dog wanting to do more. That's when they're most likely to start offering the behavior to you. And when they offer the behavior, you PAY them for that - LOL! Especially when they're first learning the trick or behavior.
To train the down, I put her on the desk so we could be eye level, asked her to sit, then held the treat between my fingers, held it to her nose then drew my hand down to the desk straight down from her nose and held it there. You don't want the treat to be sticking out enough for the dog to get it. You want them to follow the scent with their nose, try to get it out of your fingers and in the process for their body to ease down. As their body lays down, quietly praise and give them a different treat than the one in your hand.
Ask them to stand up - treat for the stand - then do it again. Keep the sessions short and always stop on a success!
Some dogs catch on to this right away, others don't. So don't beat yourself up if yours doesn't. Just get creative and figure out another way to teach what you're wanting them to learn
Also, all dogs eventually lay down. When they do, QUIETLY praise and give a treat, naming what they did ("lay down" or "what a good down" - whatever phrase you're going to use). If you're real upbeat about the praise, they'll pop right back up and that's not what you want. Only pay for the behavior you want. This is called "capturing a behavior".
If you have another dog that knows the command, you can work with both dogs at the same time. This is what I did a few times. I asked both dogs to sit and then lay down. I practice never giving a command more than one time.
My other dog, Sox, has a lightning fast down - lol! Spice (my Chi) seems to always want to know "what's the purpose here?" - hahaha! So when Sox would "lay down", I would party and pay (praised her, gave her extra treats, scratched the base of her tail). If Spice was still sitting there watching, I would say,"Too bad Spice didn't lay down - now Sox gets Spice's treat!" and I would give Sox two treats, saying,"Here's Sox's treat for lay down and here's Spice's treat for lay down!" Spice would get all antsy and jump around - do that cute little Chi dance - but I would ignore her and continue to praise Sox for her "lay down".
Then I would ask them both to stand, treat for the stand, and then do it again. It didn't take Spice long to realize that she was missing out - LOL! Now she does a great "lay down" and we're practicing the "stay".
I've also taught her to:
open (use her nose/paws to get in a door)
push (use her nose/paws to push things like a door or drawers)
touch
stand
wait
here (comes to me and touches her nose to my hand - I use this as a "check-in" then I release her to go back to what she was doing)
come (I use a special whistle and they come running)
shake hands
dance
say please ( what some call the begging position)
load up (get in the car)
My feeling is, Chihuahuas ARE tiny and cute, but they are still DOGS. Treat them like they are and you'll have your very own NPD (near perfect dog).
So in training, don't be afraid to try various methods - see what works for your dog. They're individuals and should be treated as such.
Most important, keep on having fun!