I want to build a lab to play around with RouterOS and networking setups in general.
I have an existing router/switch for my home to use. In general, how does one go about configuring a RouterOS router such that I can connect it to my existing switch to get out to the internet?
Just connect your new MikroTik router's WAN interface to your current internet router's LAN (or to a switch connected to the LAN), tell the MikroTik to get it's Internet address via DHCP (your current internet router will stand in for the ISP of the MikroTik) and make sure you configure the new MikroTik router's LAN to be a different subnet than your existing internet router's LAN subnet. After you do this, everything should just "work". The MikroTik will get its "internet connection" from the upstream router, and there will be two stages of NAT going on, but you should be able to access the internet from a PC connected to the LAN interface of your new "lab" router.
If your existing internet router isn't a MikroTik, the new MikroTik will probably work with the default LAN address of 192.168.88.1/24, since your existing router probably uses a LAN address of 192.168.1.1/24 or 192.168.0.1/24 and will give out addresses to dhcp clients from the same subnet. You can use any rfc1918 address you want for the MikroTik's LAN, just make sure it does not overlap your current router's LAN subnet.
Toasty Answer's has some "white board" videos on putting a rourter behind another router. For example these two
What is Double NAT? and
Edgerouter with Existing Network (but the same rules apply to any router, including MikroTik ROS)