The city centre is a great hub to include in any shopping trip. It's near the beach, there are plenty of restaurants, it serves as a central departure point for buses and is only 10-15 minutes walk from the station!
The major shopping attraction in the city centre is going to be Churchill Square, open 7 days a week, with more than 80 stores all in one place, alongside 2 car parks (though on busy days these fill up quick)! If you're not driving, buses to and from most of the city run along Western Road, just aross from Churchill Square.
Fashion and accessories make the single biggest appearance in Churchill Square, with over 30 of the stores selling clothes or shoes and another 10 selling accessories or jewellery. This isn't to say the rest isn't covered though! At one end of the centre, Debanhams spans all floors of the mall and sells everything from homeware, to clothes, to perfume and make-up. The rest of the stores are a pretty evenly distributed amongst phones, video games, gifts, homeware and perfume, with a good number of snacking and sit-down meal places up at the top of the atrium (see more here).
We would also consider most of Western Road (the one out the front of the shopping center) and North Street (the one that runs up to the clocktower into Western Road) as part of the centre. Along the portion of Western Road nearest the mall, you'll find most clothing brands and labels that aren't already in Churchill Square have full sized stores, covering all departments. You'll also start to spot more charity shops, phone repairs and electronics stores as you move towards Hove (which is covered here). If you go down North St, you'll also find more charity shops and discount stores, along with Boots, TK Maxx and some smaller clothes labels. You can also head south into The Lanes, or north into the North Laine area (covered here and here in our other guides).