I would agree with PP that Wavertree is good place to start, but not that house in all honesty - that part of Wavertree isn't great. I think Aigburth is a good shout, if you can get close to Lark Lane you'll find lots to do and lots of people that like/do the same things as you.
This is in a nice area: www.rightmove.co.uk/properties/112244822#/?channel=RES_BUY
www.rightmove.co.uk/properties/117939329#/?channel=RES_BUY
Also the one up thread on Errol St.
You could also look around Greenbank, and the bits of Wavertree bordering it:
www.rightmove.co.uk/properties/118893452#/?channel=RES_BUY
www.rightmove.co.uk/properties/118799936#/?channel=RES_BUY
The terraced housing stock in south Liverpool doesn't normally have much in the way of gardens but you can get close to Greenbank or Sefton Park for your budget