britrail passes to Canterbury, Warwick Castle, and Portsmouth

Different stations serve different directions of travel, which is your main concern, not the railway operator. When requesting a train schedule either through www.nationalrail.co.uk or through www.bahn.com, one tactic is to simply request LONDON or London (all stations). The resulting schedules will suggest the starting station and focus on the fastest connections.
Tips on which station to use are also in the start or end (connections section) of the relevant chapters of Rick's guidebooks.

Direct trains to Warwick leave from London Marlybone station, direct trains to Portsmouth leave from London Waterloo, and options to Canterbury East or West stations leave from multiple London stations. So, from London to Canterbury, you can request from the station nearest you. Results may direct you to take local transport to a different station.

Have you already bought the Britrail Passes? They may not be worth using for these journeys. Or maybe you have other travel planned in the UK? It would be helpful to us giving advice to know these details.

We use the National Rail website to find departure stations and schedules, as well as fares.

http://www.nationalrail.co.uk

Just enter “London” as your departure station, and it will list London, all stations as a option. Choose that.

I used June 8 as a random date, and found the following:

London to Canterbury and back (day trip, I assumed):

You can go from several stations, St. Pancras, Charing Cross, and Victoria. Travel times and prices differ. St. Pancras is the most convenient, with a no-change journey of under 1 hour, and a price of 40 GBP for a day return ticket. Charing Cross and Victoria Station departures are a bit cheaper for a day return (34 GBP) but take longer (1.5 to 2 hours).

London to Warwick, also a day trip:

Marlebone is your London station. You can buy Advance tickets for the June date for as low as 5.50 GBP each way, for a total of 11 pounds. A day return ticket, good anytime, will cost more (30 GBP) but are good on any train. Advance tickets are only good on the specific train chosen.

London to Southampton:

I assumed this is to meet a cruise so a one-way trip. You can go from Waterloo or Victoria Station.

Trains from Waterloo take 1.5 to 2 hours and offer an advance fare of 9.10 GBP or an Anytime fare of 37.10.

Trains from Victoria take 2 hours and cost 30.40. No Advance fares were offered.

I looked up the costs in hopes that you have not already purchased the BritRail Pass. But if you have, the information is still useful as to the stations. Check the website yourself and learn how to find the stations and schedules. It is very user-friendly.

For Canterbury I would leave from London Victoria and go to Canterbury East. After arriving at Canterbury East exit the station, cross the street and right there is Canterbury E Station Bridge walk up it, you will get to the Roman Wall turn right and it will take you to the city center, it’s a 10 minute walk.