I personally prefer to thank them, reiterate my contact info, and indicate the response I hope for in the last paragraph, e.g.:
"Thank you for taking the time to review my resume and consider me for
(the role). You can contact me with any questions you may have at
firstname.lastname@example.org or on my mobile at 555-555-1212. I am looking
forward to speaking with (company) about how I can help (the
department or project) succeed during a phone-screen or on-site
I try to make it less generic, of course, but that's the general format I use. Basically, I'm outlining their next steps and putting the information they need for those steps right in front of them so they can follow up without a second thought.
It's a similar idea, except you are empowering them to take the action you want from them instead of taking control directly - which can feel pushy or manipulative. Influence and empowerment is almost always better than control or manipulation in the long run.
If you don't hear from them in a couple of days, a quick "wanted to be sure you had received my application" email is an entirely appropriate follow-up, even without telling them that you will be following up first. If you do end up needing to write a follow-up note, you absolutely should slip in a line like "I really think my (skills and talents that are relevant to the job) would be great for (the company), and want to make sure my application didn't get lost or submitted incorrectly." You can also throw in something about why you want to work at that company - mention some company values or exciting projects to show that you've done your research and are really interested in them specifically. However, I've found that I've never had to do this. These days, the interview process is usually semi-automated so that applications don't slip through the cracks as much.