Getting to lodging in Heidelberg with luggage (no car)

Generally check Flixbus and Deutsche Bahn (train) connections.

VRN is local public transport provider. See network map for city of Heidelberg.

The lower funicular starts at Kornplatz, which according to Google maps is about 1.8 miles from the Heidelberg train station. Under the “fahrplan” section they suggest Bus #36 to reach Kornplatz, but do not say if that bus serves the train station. You could probably take a taxi from right outside the station if the bus doesn’t work.

https://www.bergbahn-heidelberg.de/welcome

The main train station is some distance from the old town, more than you want to walk, with a second station Heidelberg Altstadt, closer, but not by much.

There are both trams and buses, when we visited on a day trip, we just used Google Maps to find the best bus or tram to get to the old town.

You want to ride the Bergbahn to the top, right?

You will want to arrive at the  Rathaus/Bergbahn, Heidelberg stop to ride the Bergbahn.

Use the DB site to find all the connections between FRA airport and the Rathaus/Bergbahn, Heidelberg stop. Just use a date of Dec. 14 to approximate the options that will exist next year. Later connections to this stop are not yet available. Then recheck it next month.

https://int.bahn.de/en/

The resulting journeys will typically show you reaching Heidelberg Hbf by train or bus, often with a connection on the way, and then using Bus #20 from Heidelberg Hbf to the above-named stop as your final means of transport to the bottom of the Bergbahn. This bus runs roughly 3 times per hour and takes 10 minutes to reach your stop.

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If you want to stay up there, and haul all that luggage, take a taxi.

It's about a 1000 yards from the Altstadt train stop to the Bergbahn. Slightly uphill. You can walk it, and you won't be the only person with baggage in the Altstadt, but you may be fighting crowds. (From now to New Years that's a guarantee, as you have to walk straight through the Christmas Market.)

The best option would be to stay down lower (along the river is nice) and take the train to the top when you want to wander around.

Either consider reducing your luggage or do what KGC said with changing hotel or taking a taxi.
A taxi is best if you have a lot of luggage. Do you really want to stay up on that hill for multiple days with the need to take the funicular every time you want to go somewhere? The lines can be long sometimes, in both directions and you often do not get a seat.