I like to uses this schematic map of the central London bus routes to get the big picture:
https://content.tfl.gov.uk/bus-route-maps/key-bus-routes-in-central-london.pdf
You will see that 2 buses serve Paddington Station itself. Of those the # 23 is useful for getting to Trafalgar Square if you have sightseeing plans in that area.
If you walk 10 minutes down to Hyde Park, Lancaster Gate area, you can catch the 148 which runs along Bayswater Road and then heads to Westminster Abbey, passing Buckingham Palace on the way. The Westminster Abbey stop is a short walk from the Churchill War Rooms, which are right by Horse Guards Parade. After this stop the 148 crosses the river and may get you close to the imperial War Museum, but I cannot tell from this map.
Once you understand the overall route plan and know the bus number, you can use TFL and the spider maps to see real-time departures and find the actual bus stop for your bus. Most locations have different stops for different bus numbers, and they are indicated by round signs with a letter—-C, or R, or S, for example. That will be specific to the bus you want AND the direction of travel.