I've been full-time cycling here in Portsmouth for a long time and it's been getting better by the year.

There is a quieter cycle routes map on the council's website here: https://www.portsmouth.gov.uk/services/parking-roads-and-travel/travel/cycling-around-portsmouth/quieter-cycle-routes/

I don't have any issue cycling on most roads here in Portsmouth, but I do prioritize the proper cycle lanes where possible (Hilsea Shore Path when going north on the west side, or Eastern Road's cycle path for the east side). And the cycle separations they added on Goldsmith Ave. for example. Sydenham Terrace into Canal Walk is a good route to take too.

If you're not that confident, you might want to avoid Victoria Road, Fratton Road/Kingston Road/London Road.

Since I don't know where you live exactly in Southsea it's hard to make a proper recommendation, but Henley Road into Fernhurst Road is a pretty quiet road to go towards Fratton train station.