You can use actions to automate things.
Basically it works like this:
- Open the Actions panel.
- Click the Create new set button to create a new set to contain your action(s), give it a name and click OK.
- Click the Create new action button to create a new action, give it a name and click Record. Now Photoshop records everything you do. There are lots of things to consider and lots of quirks. You can experiment for yourself and read more about actions online.
- Click the Stop playing/recording button to end your recording.
- Now you can play back the same steps on another image (providing that it's actually possible).
But in your case (if you tell the whole story) I would do things it a little differently.
The background removal can probably not be automatized. You just have to find the most effective way to do it.
After doing that I would do an Image > Trim and trim based on Transparent Pixels. (You could make an action of this single step.)
After this you resize the image you say. Can I ask why you need to do that? Unless the images are in low resolution and you scale the images up to camouflage pixelation there is no need to add this extra step in my opinion. You can just scale the images in your design application and leave the images as they are.
In Illustrator you can select your placed image, open the Links panel, click the "sandwich" menu in the top right corner, select Placement Options and set Preserve to Proportions (Fit). This way when you relink to another image which might have different dimensions, it will always fit neatly into the same area.