The Box museum is in the centre of the city and has some nice interactive exhibits along with a really cool model of a Woolly Mammoth.
National Aquarium is just off the Barbican which makes a nice walk although little kids might find it a bit boring (the Barbican not the Aquarium).
There is a Clay Art shop there where your kids can paint a premade clay item and then you come back another day to collect it, or get them to post your thing to you.
Buckfast Butterflies and Dartmoor Otter Sanctuary are a short drive away (you can check their website for details).
It's just over the road from the South Devon Railway station which does special train journeys, so you might be able to time a visit with something like the Polar Express trips they are doing in December if your kids would be up for it, or the Festive Ploughmans lunch trips they do if your kids would like a trip on a steam engine.
About a mile from there is Pennywell Farm who run Christmas things some days.
There will be a Christmas Market in the city centre, with the same stuff for sale you see in every city centre in wooden huts.
They usually have some rides and random entertainment going on in the middle of it at the same time.
There is usually a city centre land train that runs up and down the main street which is supposed to be running this year, but I don't know if it will be for sure.
You want to avoid the city centre as much as possible.
Plymouth City Council decided to revamp it, so one half has recently been unveiled as a concrete wasteland after being a building site for a couple of years, while the rest of it is still a building site where there used to be beautiful mature trees.
Do not allow your children to eat outside in the City Centre as seagulls will swoop down to snatch food from your childs hands.
Last time my son forgot that, he ended up getting cut by the beak of the seagull that stole his food - they will even go for little ones in pushchairs so eat inside.
Soapbox Childrens Theatre are doing a Fairy Trail (see their website stiltskin.org.uk)
There is a trampoline park in Plymouth too, but it's for over 3s as far as I know.
It really depends on which days you will be there and how much the 2 year old will be involved in activities.
It is 99% likely that it will be raining and that there will be strong breezes coming off the sea, so make sure you have waterproofs.
There are plenty of parks and green spaces to run around in as long as you don't mind the rain.