It really depends on what you want to do (and what Teensy you have), and what your soldering skills are.
I go back and forth of how to mount stuff on Teensys, so perhaps I am not the best person to answer.
It depends, are you going to mount the Teensy permanently to the board, or are you going to use header pins? In general, I've been moving to a strategy where I have the Teensy with the outer pins (28 pins for the Teensy 3.2/LC or 48 pins for the Teensy 3.5/3.6) with a stacking header. The inner pins, I use normal female headers, but nothing that sticks out below the Teensy. This will allow you to mount the Teensy in a standard breadboard, and use jumper wires.
As the design gets firmer, I tend to think about making an under board with the 28/48 female header pins that Teensy will fit in, and then bring out all of the connections to other areas on the PCB. That way if plans change, I can just pop out the Teensy, and solder up a new board with different wiring. Usually for the top, I will think about Teensy specific boards like the Feather adapter, prop shield, etc.
I tend to like the prototype boards from:
In particular, for small builds with the LC or the 3.2, I like the 1x2" board, which is just big enough for a Teensy, and it has the 18 pin middle row to bring stuff out:
For larger builds, and stuff with the 3.5/3.6, I kind of like the mint-tin board. The Teensy 3.5/3.6 can down the middle, and you have 24 sets of pins on either side that you can run wires to bring out connections together (and you can use the Altoids tin as a box, providing the board is high enough off of the bottom that you won't get short circuits)I :
I also like the DIP-40 and the DIP-28 boards (DIP-40 is just the right size for mounting a Teensy 3.5/3.6, and DIP-28 is just the right size for a Teensy 3.2/LC):
And the 2x4" board:
Adafruit also has some interesting perma-proto boards. The mini-mint tin perma-proto (Altoids Smalls sized) is nice for a 3.2/LC with just a few attachments:
Adafruit has its own regular sized mint-tin board, plus a blank tin box if you don't want to use a brand name mint box:
And they have quarter (15 pins), half (30 pins) and full (60 pins) sized perma-proto boards:
I tend to buy headers in bulk. I will usually buy a bunch in various sizes, and when I notice I'm getting low, I will order another batch. For smaller pinouts (2-8 pins), I prefer to buy the pinouts for an exact size in both straight and right angle forms. For larger sets of pins, I will buy them in 40 pin versions, and cut them down to suit. The company Tayda Electronics tends to have a good supply:
In particular for the 40 pin variants:
Unfortuantely Tayda only carries stacking headers up to 16 pins. But you can get 18, 20, or 40 pin stacking headers from dipmicro or 36 pin stacking headers from Adafruit:
For Teensy 3.2/LC, I liked the boards from robot50.net, but the designer has stopped selling them. But she has released the PCB files so you have a PCB foundary print some up for you (as long as you keep the robots50.net silk screening).
Velleman and Busboard make some interesting prototype boards, but compared the the Adafruit boards and Dr. Azzy boards above, these boards are only single sided, so you have to do all soldering on the side with the copper. Also, it is easier to de-solder stuff in the above boards than these single sided boards.
In fact, I was just at my local electronics store (youdoitelectronics.com in Needham, MA, USA), and I picked up 3 copies of the Velleman BB02 board, which is big enough to mount a 3.6 on one side, and it has 4-5 bus lines to provide VIN, 3.3v, Analog ground, and normal ground for all of the pins.
I would suggest getting a supply of both solid and stranded 22 gauge wire with different colors. Solid wire is useful for soldering to prototype boards, while stranded wire is best for wire assemblies to move off of the board.
The company pololu.com provides headers and either pre-crimped wires or supplies for crimping your own wires so you can make custom cables. I tend to prefer the pre-crimped wires, because I find the wires I've made myself keep coming out of the housing, and I continually have to check that all wires are connecting:
In addition to standard Teensy stuff, Adafruit sells a Teensy 3.2/LC -> feather adapter, that allows you to mount the various Adafruit feather wings (Adafruit name for shields on the feather platform) on a Teensy 3.2/LC. You can mount the feather board on a 3.5/3.6 as well, but it takes a bit more work.
Besides using feather boards, the feather adapter has a lipo battery port, and it can charge the battery if you have the USB port connected, and provide power to the Teensy if the USB is not connected.
Here is a little project I did some time ago, that combines: