Oh dear, don't go near that town centre. Chav central.
Parts of Southsea are ok (still too many chavs though).
No decent riding without a drive, unless you take the "move to Chichester" advice.
I live in Waterlooville, which isn't far from Petersfield where the Downs start. It's ok. some resonable roads for quiet roadying, if you like that sort of thing.
Too many cars in Portsmouth too. It has to be one of the worst cities in the country to drive into, or even ride into. Traffic is horrendous. So is the standard of driving. And parking? Don't get me started!
Anyway, yeah, look at the surrounding area, where there are some ok spots. I've lived in a lot of them!
If you find somewhere interesting and want to ask about the area, drop me a message (or post on here).