Multiple use policy of of BLM land that includes grazing leases generally is open to prospecting. The problem often seems to be having to cross private property to get to an unclaimed section of public land. If that is the case, the property owner can restrict access to his land to prevent access to the public land. Ranchers with grazing leases can not exclude other multiple use participants such as campers, prospecting etc but there may be existing laws which govern how and where such use can be exercised, such as proximity to water holes, wildlife and so on.
In General, prospecting on UNCLAIMED BLM and National Forest land is lawful (1872 Mining Act), and allowed, (subject to applicable rules and regulations of those jurisdictional agencies) However, State land (State Trust Land) is closed to mineral entry, meaning no prospecting allowed. There may be individual exceptions to prospecting on State Trust Property (so I'm told) but a special application procedure is required which is costly and involves a somewhat lengthy process. (details may be obtained by contacting the agency of interest)
You need to learn the rules from the BLM. It is a bit more complicated than Skip presents it. In short if it is not claimed and there are no other restrictions then yes, you can pan and do small sampling.
State land is not public land like BLM. State land is private land and you generally can't prospect at all. Different States have different rules but the ones I am familiar with do not allow prospecting.
You asked a huge question. One that requires a lot more than a simple answer. A little research on these threads and some time spent on the subject is the only proper way to answer the question.
Finding a spot is the key to finding gold. The land research and laws are as steep a learning curve for new guys as the prospecting techniques.
Here is my best advice to a newbie. Don't look for "open ground" because you aren't going to find squat. Look for the most tightly held claims in the district and get as close as you can to that. It will be a pantload easier to get on real gold by meeting people and learning who is who in a district than trying to learn to prospect in an "open" area.
I often ask forgiveness instead of permission. I have also noticed that the gold is a lot brighter on corporate land. And you will find many ounces of gold before you will find "open" ground worth working.
Just my two cents.
If the area is opened to mineral exploration, and not already claimed for either a lode or placer claim or any other mineral claim, you can do any kind of prospecting your heart desires, i.e. metal detecting, dry washing, sluicing, panning, magnetic imaging, divining rods, and even psychic readings, etc.