I asked a similar question on these forums a few weeks ago.
I ended up removing half the slurry from batch #1 and just pouring batch #2 straight on top. Cleaned/sanitised the lid and airlock but not the fermenter.
This was a massive over-pitch, fermentation started within hours and blasted at full noise for a few days with huge krausen. Hit FG very quick. I should have removed all but about 1 pint / 500ml of slurry (for 23L/6G batch), will do that next time, but no harm done. The second beer came out amazing, I'm super pleased with the taste.
My advice, FWIW, is to keep it simple and pitch onto the cake rather than washing first. Unless the first beer was super hoppy or high gravity or something special as already noted. This way it's very simple, less clean up, less handling so less risk of infection.
With the yeast that you remove from the fermentor, wash that and store that in the fridge. That way you get some practice at washing yeast without risking a batch of beer if you screw it up.