Challenges in Research

Funding

Some granting agencies only make their funds available to U.S. citizens or permanent residents.

In some cases those policies are negotiable. Talk to the program officer and to your institution's Office of Research to discuss the possibilities.
Many agencies (including the National Science Foundation) do not have such restrictions. Search them out.

After you have applied for lawful permanent resident status, you aren't supposed to leave the U.S. until your case is decided.

If you have a compelling reason to travel, you can apply for what's called "advance parole." Such reasons might include the need to attend or present your research at an international conference, or to do field work abroad, either of which could be essential to your professional advancement. Note: you must apply for, and receive, advance parole prior to traveling abroad. Otherwise, you may not be able to re-enter the U.S. Because the laws governing citizenship and immigration are subject to change, you should verify this information with the U.S. Citizenship and Immigration Services.
Citizens of many countries need a visa to travel abroad, even if they are lawful permanent residents of the US. Apply for a visa as soon as you know your travel plans.

This can be challenging for any new faculty member. Successful strategies include:

Network with colleagues who teach undergraduate courses in your specialty.
Go to professional conferences and keep an eye out for promising undergraduate students with interests similar to yours.
Serve on the graduate admissions committee for your department.
If there is a required course for incoming graduate students, volunteer to teach it, and use examples from your research in class.