Depending on how much you plan to walk each day, I'd start in either Burgos or Leon. They are large cities, with nice albergues, and you're likely to meet up with some other pilgrims to walk with. If you start in Leon, and walk fast, I'd also recommend walking through to Finisterre. It is well worth the three additional days on the trail.
With three weeks, Burgos at 485KMs from Santiago is almost 25kms / day and Leon at 305KMs is about 15Kms / day. Most people would probably opt from Burgos with that amount of time. Enjoy the open plains of Castile and start from Burgos!
There are many choices. I just completed the Camino Frances on Sept 30.
If I were you I would go from St Jean Pied de Port to Burgos. I would then take the bus to Leon and would go to Santiago from there. You avoid the Meseta and the Paramo which are furnaces in the summer and fall (I have done it...)
Peaceable Projects Inc. is a U.S.-based non-profit group that brings the vast resources of the wide world together with the ongoing needs of the people who live, work, and travel on the Camino de Santiago pilgrim trail network in Spain.
Actually, I am walking from Burgos to Finisterre and it's taking exactly three weeks. I walked the same distance two years ago in this amount of time, with some different stops, but these are the planned stops for this winter. Here's my walking schedule:
Burgos to Hontanas 29 KM
Hontanas to Boadilla del Camino 29 KM
Boadilla to Carrion de los Condes 25 KM
Carrion to Terradillos de Templarios 26 KM
Terradillos to Calzadilla de Hermanillos 27 KM
Calzadilla to Mansilla de Mulas (about) 27 - 30 KM
Mansilla to Leon 19 KM (short day, explore Leon)
Leon to Hospital de Orbigo 31 KM
Hospital to Astorga 17 KM
Astorga to Manjarin 32 KM
Manjarin to Ponferrada 23 KM
Ponferrada to Villafranca del Bierzo 22.5 KM
Villafranca to O Cebreiro 31 KM
O Cebreiro to Samos 32 KM
Samos to Portomarin 32 KM
Portomarin to Palas de Rei 26.5 KM
Palas de Rei to Arzua 29 KM
Arzua to Santiago de Compostela 40 KM
Misa del Peregrino @ 12:00 then Santiago to Negreira 22 KM
Negreira to Olveiroa 33 KM
Olveiroa to Finisterre 35 KM