What parts do I need to hook both units to this sink?
Assuming you can physically fit a dishwasher and a washing machine between your cabinets, you would need to replace two parts.
- Replace the trap under the sink with one that has connections for two appliances.
- Replace the blue shut off valve with one that also has two connections.
Here are a couple of parts that should work.
This is all feasible by a DIY'er but might require some extra fittings or cuts to the shelf to make the new trap fit. You might also be able to find a similar valve but push-fit and easier to install.
You can not add a washing machine to this connection. You can add a dishwasher with a Plastic Dishwasher Branch Tailpiece
You may have to cut this to size, though, as it looks like you have limited room between your sink bottom and your in-wall plumbing.