You're not going to find any useful weather forecasts for 3 weeks out.
https://aviationweather.gov/progchart/sfc?fore=thumb
However, starting a week out, you can start following the prog charts to get an idea of how things are trending and planning. If you see a 3 day window of good wx in the region you are planning (in other words trip day +/- 1 a day) you're in pretty good shape. If it's iffy, you'll have to wait for more info closer to.
https://aviationweather.gov/windtemp/plot
Starting about 3 days out, particularly in winter, you'll want to start watching the temps aloft as well. Select Plot > Temperature, Vert Level > Cruising Altitude, Time > Departure/Arrival. This plots the forecast temperatures aloft around the entire country. Can help with thinking if icing is a concern or not.
Lastly, it's not the weather that you're going to be planning 3 weeks, 3 months, or 3 years out. It comes down to knowing how to plan trips in a private airplane to account for the unexpected. And weather is not the only concern. Maintenance problems, pilot health issues, etc.
I've done a lot of trips around the country and nearby countries and have found that short of major maintenance issues, I can make just about any trip in +/- a day. Heck, usually it's +/- morning vs evening of the same day with IFR capability. As the more accurate 3 and 1 day out forecasts start to come in, I start planning a contingency. If the day of my trip is looking impossible, I might leave a day early to make sure I make it on time. Worst case, I usually plan a low key day the day after arrival just in case I arrive late. So in other words, never plan an important business meeting on the day of arrival or the day after arrival. Have that buffer of coming a day early or day late. That covers the vast majority of incidentals. Even things like needing a starter or a tire, not just weather.
I think of this strategy not only as a weather strategy but also largely as a risk mitigation strategy. Many accidents are the result of getthereitis. By planning a 3 day window for getting places, it greatly reduces the pressure to perform a flight or continue a flight when you should not. Even with the best of advance weather planning, what you actually get can be quite different. There's always the possibility of having to cut it short or divert. So, having reserve time planned is important.
With that said, the majority of flights go smooth and as planned. Usually there's a way to divert around weather (if it's not at the departure or destination) or alter the time a little earlier or later to make it work. The spare days relieve the pressure and help deal with more severe weather. And finally, when that plan isn't good enough, it's time to start thinking about other modes of transportation. Particularly the train or greyhound because even the airlines can be affected by weather. The kind of weather that will ground us for multiple days in a row (hurricane, snowstorm) is probably going to ground them too!
