St. Lazare to the Gare du Nord

Cheapest is almost certainly the bus or Metro. I am sure I have made this connection in the past but don't recall - today I'd just go on Google Maps and look it up. The bus might be direct so that would probably be easiest with luggage. Some people hate public transportation so for them, neither bus nor Metro would work for them.

Cheapest is to walk. It takes about 30 minutes. Easiest is a taxi.

I took a taxi from Gare Du Nord to St. Lazare in October. It cost about 9 Euro.

By public transport, rather than the metro, you can take the RER E suburban line just for one stop, to Magenta. There are escalators everywhere so it is not that bad with luggage.
A taxi is a little bit easier, but can get stuck in traffic, especially with the current roadworks on the direct street tbetween the two stations (rue La Fayette).