You do NOT need a permit to leaflet, circulate petitions, pass out information, hold signs, request donations, or otherwise express political or other opinions, peacefully and without obstructing traffic, in public spaces such as sidewalks or parks. By doing so, you are affirming your (and everyone's) rights to free speech and assembly.
If approached/confronted by a police officer or security guard, explain that your rights are protected by the US Constitution, and that you are not interfering with others' rights (freedom of movement, etc.). You do NOT have to answer their questions. You CAN protest vocally and loudly if you are being shoved, grabbed or otherwise touched, but do NOT push or fight back. Call “cameras” to bring others close to the incident to photograph, livestream and/or take video of the incident.
If the harassment continues, call an attorney or other support people, and document the incident with still photos, video, audio and/or written accounts.
Never touch a police officer or his/her equipment. If arrested, do not physically resist arrest, although you may decide to call out your protest of the arrest, especially if it happens for no apparent or good reason. For more info and support, contact the National Lawyers Guild or the ACLU.
Take a self-defense course to learn/relearn/practice skills in situational awareness, and physical presence and movement.
Read the Security Culture guide from The Ruckus Society.
Especially in high-intensity actions, but in general, team up with one or more action buddies. It's safer and more fun. Show up and leave together - or make a plan if one of you needs to leave before the other. Make sure you have each other's cell phone numbers and that your own cellphone is charged and where you can easily access it (in a pocket rather than somewhere in a big backpack, for example).
Attend a nonviolent direct action training. You'll be able to role play, discuss issues and meet others in your community acting in resistance to injustice.