The easiest slip is at Itchenor:
Accessible at all states of the tide. Most (all?) the other slips are limited to about 2 hours either side of high tide, except for the expensive one in Northney Marina. The only issue with Itchenor is that the road down to it can get very clogged with holiday makers.
There is a bit of info about angling in the harbour here:
From Itchenor, you can head down to to Bosham or Emsworth - there is a visitor pontoon at Emsworth and a quay at Bosham - you need to tie up at the nearest end of the quay at Bosham to avoid charges.
There is a chart of the harbour here:
If heading out of the harbour, be aware that conditions across Chichester bar can be very nasty.