Reading this thread has inspired my curiosity in to looking at Zwift myself - I found some useful web pages that I've linked to. I've also had a quick look at the Clarke CCTI Italian Bike Trainer and it is a classic (dumb) trainer similar to my Elite Chronofluid.
How to get started on Zwift with a classic trainer
Supported trainers
The important issue here is that your trainer is not one of the ones supported by Zwift, by that I mean they have no data for your trainer to use within their app regarding power levels (it could make you look like Chris Froome and it will upset the natives

) so something like Zwift might not be an option unless you invested in an actual power meter for the bike, a Zwift compatible smart trainer or a Classic trainer that is on the Zwift list. These will cost more than your original trainer.
Zwift Set Up
You could get something like a Garmin 500 with a speed sensor and heart rate strap. This is the setup I currently use with my trainer - although I'm not on Zwift (I would need an ANT + dongle to enable my Garmin 500 to connect to my PC to enable me to do this) .
However I can see my stats whilst I'm 'riding' and record these stats (speed, cadence, heart rate, distance, time etc) and upload it to Strava to have a look at later - other sites may be available.
Garmin 500s can be had pretty cheaply now as it's very much old tech (2nd hand from
ebay is a starting point) . Choose an ANT + compatible heart rate strap (not essential unless you want to know your heart rate) and a speed sensor (essential, otherwise you won't be able to track your 'speed' on the trainer) - they don't need to be Garmin Branded and they will be much cheaper if they are not.
The other option which has already been suggested is to keep it really simple which has a lot going for it unless you want to start spending.
Because your bike will be stationery, just having an App on your phone won't be any good because there is no movement to measure. You need some additional gizmo's attached to your bike, which measure movement on the rear wheel (as that will be the only one that's going around).
Depends how much you want to spend.
This one at about £25 should be able to measure basic speed and distance:
https://www.wiggle.co.uk/cateye-velo-wireless-plus-cycle-computer/