How long did it take for Frodo and Sam to get from the Shire to Mount Doom in Mordor and drop the Ring in?
Frodo and Sam left Bag End the day after Frodo and Bilbo's birthday, September 23 3018 TA (exactly 17 years after the night of Bilbo's disappearance). The destruction of the Ring at Mt Doom took place March 25 3019. So the entire journey took about 6 months. But note that this included about 2 months spent in Rivendell, and a month in Lothlorien. See the LotR Timeline (lordotrings.com) and Timeline of Arda (Wikipedia) for more details. Many of the dates can also be found in the appendices of LotR.