We would like to spend a few months in an apartment rental, as a home base, centrally located so we tour different countries. Easy train access.
Not sure where to start. Any ideas?

Latvia, obviously.

Depending on how far afield you plan to wander, is a single base the most practical?

Need more information on what you want to see and how long you want to spend in various places.

For example, Paris is a good base for many day trips but are you prepared to essentially pay double overnight costs if you plan to spend a week in Italy? That doesn't seem very practical from either a cost or time perspective.

Better, perhaps, to choose 2-4 bases depending on length of stay.

When? How many people? How much time do you have? Are you aware of 90 day rule? What countries interest you? Where have you traveled already?
Agree with Cathinjoetown. Europe is so big, one base doesn't work well unless you are willing to leave it for a week or so at the time, paying for both the apartment and wherever you stay while traveling.

Remember without an extended visa you are limited to 90 days in any 180 day period.

Unless you choose a base in the UK (which allows up to six months) near an airport with cheap flights across Europe.

"Remember without an extended visa you are limited to 90 days in any 180 day period."

Assuming the poster is North American or Australian, true only for the xenophobic countries in the Schengen.

I know this will come as a shock to EUfanatics - but there are other countries in Europe.

That's rich coming from Brexit Britain where the vote for leave was based almost entirely on immigration by Europeans.

The US has a similar restriction on visitors to it's shores, even Britons, who seem happy to have Americans there for half the year.

Many people are unaware of the Schengen limit so it is reasonable it is stated early in the piece, though it could have been made clearer that it only applies to Schengen signatories.

Thanks for the info. This will give us a starting point.

Germany is fairly central, or Switzerland. Both have good train systems.