If you milled or etched your own board, it won't have plated through vias. Put a small bit of wire in each via and solder both sides of the board.

Use some perfboard to make sure the pins are straight, and double check by putting the processor on top of that. The right angled pins on the bottom should face away from each other. Make sure pins don't overlap PCB traces, especially if you milled your own board and have no soldermask. Solder all pins, but it will probably go easier if you solder the ends first.

You can either solder the processor in permanently, or else put female headers on it first. If you want the headers, just place them on top of the pins you just soldered and follow the instructions from there.
Place the processor on the pins in such a way that, if the speaker hole is above it, the debugging connector is on the left, and the USB connector is on the right, and the reset button and jumpers are facing you.

You can use thru hole parts or surface mount parts; pads are provided for both. Just don't use both at the same time.

If you have an XH-M125 amplified board, you can simply solder a pin header (or socket) to the provided place for the connector on the board.
Otherwise, all of the necessary surface mount parts mount on the back of the board.

They all take 2 x 1 pin angled pinheaders. The battery connectors point away from the speaker. The speaker connector points straight down.

Nothing special here.

If you are not using an external LED ring, you'll have to solder the surface mount WS2812B RGB LEDs by hand. I think it works best if you use lots of flux, get the part into place, and then hit it with solder. Be aware of the corner with the notch and matching it up with the diagram. If you spin the ring each time so that the LED you are working on is on the top, pin 3 (with the notch) will always be the upper right pin. You may also want to solder the safety capacitors at this point too.

Take an old pen and clean and hollow it. Get a non sharp, solderable bit of metal to use as the point. I used the tip of a 1/4" plug. Put your holes in your pen for the wire to get to the tip and thread the wire through. Solder the wire to the tip. Reassemble the pen. If you are using some sort of connector on the end where it attaches to stylus, attach that now.
On stylus either attach the mate for that connector, or just wire the stylus directly to the board.

The battery holders have leads coming out, but they need connectors to attach to the pin headers on the board.
The speaker has no leads, so it needs leads with a pin header socket on the end and a speaker.