Planning on flying into Bergen for a few days. Could I get a suggestion on doing the Norway in a Nutshell plan? I was thinking of doing the Sognefjord trip and staying a night in Flam. Can you do that or do you by the trips separately? Thanks!

NiN is a collection of public transport tickets that you book together in one package. So you can easily "roll your own"

So do you have to buy the tickets for specific times and days? Would you suggest taking the train or renting a car? I was thinking of spending a night in Flam.

Makes no sense doing NiN by car - it is a route on public transport and with a car you would have to take return trips on the attractions to get back to the car. With a car of course you could visit less well beaten paths and avoid Flåm altogether.
And yes, train and boat tickets are for specific departures - many sell out in high season.
The Sognefjord trip is a looong ride on an express ferry in the wide Sognefjord. The NiN is a slow ferry through the gorge like UNESCO heritage Nærøyfjord and a trip down the Stalheimskleiva switchback.

can you do the NIN from Bergen? And do you recommend it over just the Sognefjord? Just looking for some guidance on to whether we should fly to Bergen or Oslo. Thanks!
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Yes, NiN recommended over SiN for reasons given previously. NiN can be done as round trips from either Bergen or Oslo, or Between these two cities in either direction. If you mean you can fly directly from home to Bergen then do that.