There are numerous designated trails in wetlands around the country, including boardwalk routes granting you access to the heart of marshes. Such trails are great ways to get a taste for these productive ecosystems without getting your feet wet and without unduly disturbing marsh habitats. Take advantage of them.
Many more trails skirt the edges of marshes, providing plenty of opportunities to observe wetland flora and fauna.
Many marshes, however, lack trails---for obvious reasons. Check the regulations on off-trail travel with the appropriate managing agency; sometimes it's entirely prohibited, sometimes you need to file a backcountry permit. By their very definition, marshes are never particularly deep. The National Audubon Society's Wetlands defines shallow marshes as those 6 to 12 inches deep. Deep marshes are up to 6 feet deep. In areas where it's legal, it's definitely possible to slog through with rubber boots or waders---but a stick, trekking pole, or other tool is useful for estimating the depth and stability of substrate ahead of you.
In trail-less marshes that allows entry, take along a good map, a compass and a GPS. Disorientation is not uncommon in large wetland expanses. Make note of the high ground you enter a marsh from. What landmarks can you identify---a grove of tall trees, for example?
Be aware of the typical geography of whatever type of marsh you're visiting. This will help in traveling safely. For example, if you're in a sawgrass marsh in the Everglades or the Big Cypress in southwestern Florida, and you want to explore one of the "tree islands" (hardwood hammocks, cypress domes, willow heads, etc.) dotting such tracts, keep in mind that a moat of deeper water---the result of plant matter from the hammocks or heads decaying limestone bedrock---often surrounds them. Shallow marshes may fringe deeper ones or lakes, so keep aware of changes in depth.
Enjoying Marsh Flora & Fauna
Salt and freshwater marshes are some of the most ecologically productive habitats in the world. No matter where in the U.S. you're exploring a marsh, you're bound to get a sense of this diversity of life. Marsh plants stand above water, float on its surface, or grow submerged, and come in all shapes and sizes---from the familiar cattail (the shoots and taproots of which are edible) to the fragrant skunk cabbage.
In most marshes, birds are some of the most highly-visible animals; wetlands provide critical habitat for species during all parts of the year. Depending on the location and season, you may also see or hear frogs, water snakes, deer, muskrat, nutria, river otter, turtles, or American alligators.