Hi 1066. It sounds like your DS is doing really well! And I totally understand how you feel regarding teaching him your mother tongue, I haven't given up on that dream yet, I still want DS to be bilingual, I've just put it on hold for the moment because I know he couldn't cope.
Also, your situation and mine are different as DH is english and I am not, I can see why you would want to continue speaking in english at home rather than spanish.
Sorry, I hadn't seen your question before ("What sort of things do you do with him at home to support his language?")
It is really difficult to summarise, but I'll try to give you an idea. The sort of things I do with him change over the months depending on what he has learnt, but currently, I am trying to follow the ABA principles to motivate him to learn. What I am doing now is more or less the following:
Nouns: Every week I will teach him two or three new nouns (this is easy for him now but it was hell just 9 months ago) I do this by showing him pictures, using the object, finding a pretend object and playing with it, whatever I can think of.
I test to check if knows the nouns by putting plenty of objects or pictures on a table and asking him to hand me xxx, or point at xxx (making sure not to give him any clues). I also put teddy and dog on the table and ask him to give xxx to dog and yyy to dog. I will then put objects in one room and while we are in the kitchen for example I'll ask him to please get me xxx. I have now started teaching him questions with "WHICH" so I also use the function of the object to see if he has understood, for example, show him a pineapple and a t-shirt and ask him which one do we eat, or picture of a helicopter and car and ask him which one flies,etc.
Every few weeks I start a new "programme", so the types of programmes I have done with him are the "who", "where", "whose". Again you have to make sure he understands, so for example: I took a picture of DH and I, DH was wearking a hat, so I asked him "Who is wearing a hat?", I took pictures of all the rooms in the house and asked him "where do we cook?",etc. When we are eating I'll ask him, "whose place is this?", where are we eating?, etc...
I am also working on me and mine. So for example we play a ball game with DH and DS and ask "Who wants the ball?" and we prompt him to say ME. (He can now do this). He understands "whose" now, so he is now able to say "mine" when needed.
He can now do sorting (classifying things by type, for example: all cars together, all animals together,etc). So on the language front what I did recently is getting him to label the actual group. For example, if I asked him to sort animals and items of clothing, he was great at saying the names of every item, and he could classify them, but he didn't actually know how to call them collectively, so I taught him to say animals, clothes, etc
Verbs: He is doing really well with verbs. I bought a set of pictures which worked really well for this. Using the ABA principles at first to start with worked well. So for example, he loves to be tickled, or to chased, so I prompted him to say tickle, and then tickled him, or to say run and then run after him, then I would show him a picture of someone running, then I would get teddy to run and tickle dog, and I would ask DS, "what is teddy doing?", I can now show him a picture and ask him "What is the girl/boy doing?" and he can reply,etc.
Adjectives: I started with big and small which is easy to demonstrate, I got a HUGE horse and a tiny horse, then showed pictures of a big car and a small car, etc... Then dirty /clean, again showed him over and over identical cloths, one dirty one clean. Took pictures of the car before cleaning/after cleaning, etc. This I find take a bit more time.
Likes/Dislikes: Get something he likes and doesn't like and get him to understand what "I like xxx" means, or if someone asks him "Do you like xxx" to answer yes or no.
Intraverbal: Lots of songs/phrases where he needs to complete the or fit in a word in the middle I found has helped as well.
Prepositions: The best way to teach DS this is by playing, for example, I tied a couple of chairs with a lose rope and taught him to throw a ball under/over the rope. Pictures also work.
It is a bit tough to write it all down, but I think you get the idea with what I have written above.
(sorry it was so long!) Good Luck!