Frankfurt in December

1. I want to take the train to Nuremberg to the Christmas market. Any advice on trains, is it possible to do in a day?

yeah, take direct train every 1/2 hour or so e.g.
Frankfurt(Main)Hbf 09:54 2:0 ICE 29,90 EUR  57,00 EUR
Nürnberg Hbf 11:5

2. I want some fun things to do in Frankfurt in December....any suggestions.....also names of cozy little restaurants for dinner

I do know but they've got big Red Light District the Rick Steves recommends in his book.

3. Any other picturesque towns accessible by train for a day trip.

Hey I was there March 2016 and ended going to Rothenburg ob de Tauber on a day trip
It takes a change of train in Wurzburg (I also stopped there to hang a bit). e.g.

Frankfurt(Main)Hbf 10:18 +0 2:31 2 ICE, RB 39,90 EUR  47,00 EUR
Rothenburg ob der Tauber 12:49 +0

1. You can actually get an ICE return ticket for as low as 39.80 EUR. Looking at Dec 6, leave Frankfurt at 8:54 and take a 9 or 9:30 pm train back to Frankfurt. A 2 hour journey. But to get those prices, you have to buy soon and you will be absolutely committed to traveling on the exact trains on the ticket.
2. Marburg might be worth a visit. 1 hour regional express train every hour. It is a 30.20 EUR RMV Tageskarte. It is a university town. It was also declared a "hospital" city in WW II, so didn't get bombed.

what is the best website to purchase train tickets on

Deutsche Bahn tickets at:
https://www.bahn.com/en/view/index.shtml

RMV tickets, (the regional transit authority) you just buy out of a vending machine.

I don't think I would do a day trip to Nuremberg for a Christmas Market when there are so many other great markets right around Frankfurt. Wasted day on a train.
There is a ton of stuff to see in Frankfurt and the RLD isn't one of them. Why Rick dotes on having people walk through here as though it is a zoo is puzzling. We have world class museums, medieval churches, fabulous Farmers Markets, meaningful and touching Holocaust Memorials, a wonderful botanical garden, and beautiful old neighborhoods.
Favorite restaurants - Atschel, Dauth Schneider, Klosterhof, Weinstubbe, Apfelwein Solzer, Zur Sonne, Malepartus.

• 23 Nov. - 23 Dec. Rüdesheim (closed 26 Nov.)
• 24 Nov. - 29 Dec. Limburg an der Lahn
• 25 Nov. - 17 Dec. Bad Homburg, weekends only (closed 26 Nov.)
• 27 Nov. - 23 Dec. Worms
• 27 Nov. - 22 Dec. Heidelberg
• 27 Nov. - 22 Dec. Hanau
• 28 Nov. - 23 Dec. Wiesbaden
• 29 Nov. - 23 Dec. Aschaffenburg
• 30 Nov. - 23 Dec. Mainz
• 30 Nov. - 17 Dec. Seligenstadt
• 28 Nov. - 23 Dec. Hanau
• 1 - 23 Dec. Marburg
• 1 - 23 Dec. Michelstadt
• 1 - 23 Dec. Darmstadt Small markets
• 30 Nov. - 3 Dec. Oberursel
• 1 - 3 Dec. Idstein
• 2 - 3 Dec. Kloster Seligenstadt
• 2 - 3 Dec. Frankfurt Höchst, Medieval Market
• 2 - 3 Dec. Ronneburg Castle, Medieval Market
• 2 - 3 Dec. Boppard, Medieval Market
• 2 - 3 Dec. Hessen Park Open Air Museum
• 6 - 10 Dec. Büdingen
• 8 - 10 Dec. Königstein
• 8 - 10 Dec. Friedberg
• 8 - 10 Dec. Ingelheim
• 9 - 10 Dec. Ronneburg Castle, Medieval Market
• 9 - 10 Dec. Wertheim, Medieval Market
• 9 - 10 Dec. Eltville
• 9 - 10 Dec. Kronberg
• 9 - 10 Dec. Bacharach
• 9 - 10 Dec. Hofheim
• 9 - 10 Dec. Bad Soden
• 9 - 10 Dec. Kloster Seligenstadt
• 9 - 10 Dec. Neu Isenburg
• 9 - 10 Dec. Schloss Vollrads, Oestrich Winkel
• 14 - 17 Dec. Gelnhausen
• 15 - 17 Dec. Bad Vilbel Historic Wasserburg
• 15 - 17 Dec. Ingelheim
• 16 - 17 Dec. Eppstein
• 16 - 17 Dec. Hessen Park Open Air Museum
• 16 - 17 Dec. Oppenheim
• 16 - 17 Dec. Kloster Seligenstadt
• 16 - 17 Dec. Ronneburg Castle, Medieval Market
• 16 - 17 Dec. Wertheim, Medieval Market
• 21 - 23 Dec. Ingelheim
• 27 Nov. - 7 Jan. Speyer