Unfortunately, it is very likely you will not be alone where ever you decide to go fishing around town. The Town of Silverthorne is generally the main "non ski resort" hub of Summit County, and the Blue River is the main river that runs right through town. Some other posts have focused on other drainages that could be an option, but I will focus on the Blue because it will be right outside your door.
For some local info (albeit from a shop) go to Cutthroat Anglers in Silverthorne, and ask for Troy. If he is there, he will generally shoot you straight about where to go and what will be working. If you want to hire a guide for a wade trip, I would ask Troy if he is available or who he would suggest out of their shop. I have no experience with Colorado Anglers because they are kind of new to town, but CA has been around in Silverthorne for a long time.
The Blue River runs right through the middle of Silverthorne, and is a tail water fishery with the release from the damn approx 1.5 miles above the CA shop. There is lots of public access along the river through town, but these areas get hammered by skills ranging from never-ever novice to the most experienced almost constantly. The fishing pressure decreases if you get in your car and drive north out of town, but be wary of private property once you get out of town (these folks call the cops). The road follows the river more/less for around 10 miles, but only stop to fish at designated public access points. Even the stretches out of town get pressure, but not quite as much as in town.
Another option is to drive 30 minutes from town to Green Mountain Reservoir, and fish the canyon area below the damn. This is also a tail water fishery, but it takes some effort to get there and down to the river. You will need to park in the designated parking area below the damn, and walk/scramble down a very steep trail to the river. This also gets some pressure, but there are some nice fish to be had. I have heard you can fish for about 3 miles downstream of the damn before getting into private property, but it "canyons out" about 1 mile below the damn and you have to cross the river (if flows aren't too high) in order to find the trail and continue downstream from the canyon part. Here are two pictures of what the Blue below Green Mountain looks like in the summer at a flow of 600 cfs +/-, you will probably have more in the range of 150cfs in March. It is a cool place to fish if you have the time to make a day trip, especially if you fish a few hours below Green Mountain and also stop at a couple spots along the Blue as you drive back to town.
The guys at Cutthroat can tell you about the options in and around town, and below Green Mountain as well. If you can fish mid-week, you might get lucky and be alone but otherwise you will probably have some company. If you will be there in March you will have to contend with weather - even some areas of the Blue may have ice blockage and bridges (like out of town further from the damn) if it has been cold and snowy. If that is the case, your only options might be to fish directly below the damns along with your new close friends who are looking for the only ice-free water too. Where ever you go, know that the fish are well educated and can be fickle due to the pressure - these areas just don't have dumb fish like you can find at more remote locations around the state. I find that nymph patterns with some flash work best (flashback PT mentioned), and if you are below Green Mountain try a BH Flash Prince as one of your choices in a double nymph rig. Have fun and stay warm.