Make sure to do some research on the various programs and research sites. Do not just apply blindly - spend some time on the internet browsing through the general web page for the university department, national laboratory or observatory you are interested in. If possible, identify certain types of research done at each program or site that interests you and make note of it. Also, try to identify aspects of these programs that U Iowa does NOT offer (see below for more on this) and identify reasons why you would like to go to one of these labs or observatories.
Keep track of application deadlines and materials - most applications (including letters of recommendation) are submitted online. If letters of recommendation are submitted by mail, then provide stamped, addressed envelopes to your letter-writers to help make sure that they get the letters done and correctly addressed. Make sure to give your letter-writers enough time to write the letters and send them off before the deadlines. It's a good idea to decide on which programs you'd like to apply to, and then to provide a summary sheet for your letter writers with the deadlines and information. Also, be sure to request or copy transcripts (sometimes photocopies are acceptable) well ahead of time so that your application packet is complete by the deadline.
Most of the summer research programs require some kind of letter or statement of intent. Most of them should be 1 page in length (some may need 1-2 pages if they request lots of additional information about coursework, etc). Keep it short and concise, but also remember to include all relevant experiences you have had doing research or in your coursework. It is also important to demonstrate that you have some particular research interests and hopefully mention that the site you are applying to offers those possibilities. Although it may take more time, it is worth your while to actually tailor each of your essays/statements to the particular sites, rather than a generic statement.
A note: some of these programs are actually quite competitive, so you want to make sure your application is well-put together and complete. Because U Iowa has a Physics and Astronomy department active in physics and research with many opportunities to get involved, you want to stress in your application that the opportunity you may get at the REU site is not one your current institution offers. That is why you may have better luck and better experiences at a national lab or observatory rather than just another university, although many of the university-site REU programs offer excellent, unique research possibilities.
If you have time, have someone (a faculty adviser, one of your letter writers) read over your application and offer their suggestions.
There are actually MANY ways to get involved with research around the department. Start talking to faculty early in the spring semester or as soon as you decide on your summer plans to have the best opportunities for funded research opportunities. There is a departmental award: the Van Allen award to help support summer research and several university-wide sponsoring agencies. Usually it's best to find a faculty member willing to work with you, and then discuss funding options. Some students work part time and take summer courses, or work full-time for a month. There are many possibilities. A summer spent here at U Iowa getting your first experience with research is often the best qualificiation you have for applying for off-campus summer research jobs.