Can I take the old sim from the old Motorola & put it in the new phone please - the smartphone?
You can.
Although you may want to check if the contract still suits you needs. You're likely to be using a bit of data with a smartphone where before you'd presumably only have been using text and minutes.
Sim cards are only tokens to represent the contract, they don't even retain the number they were on when deactivated so the one in the travel router at the moment has no value.
...as long as it fits physically.Newer phones tend to use smaller format SIMs vs. older or more basic ones.If not, it's usually just a matter of cutting it - but you may want to go to a retail store for your network to have that done - or a new SIM provided.
Thanks, I'd better go along to a shop, I am new to smartphone technology. There are (or were) several phone shops near me, are there any that are recommended please? There's a bigger town nearby, I'd have more choice. Thanks. It would be EE for me since they are the new Orange.
If your service is with EE - go to one of theirs.