It can be done with a stick, but it would be much easier to use thicker tubing for at least one side of the joins.
Gasless mig is a good cheap alternative for welding thinner steel but has some drawbacks when it gets real thin.
Mig with gas is hard to beat for speed, adjustabilty and ease of use, but has the added costs.
Tig is the ultimate, but requires a fair amount of skill, is slower but neater, and has a greater up front expense along with the gas costs.
It all depend on what you want to spend and how easy you want it to be.
I am trained in all of them and capable of building a bike with all of them, a qualified metal fabricator and have won awards for my work, my choice is the Gas equipped Mig, a good balance of ease and speed verses looks. The tig will produce a nicer result, but I find the speed of the mig allows me to accomplish more in far less time and be willing to try things I otherwise might not.
Only the Mig options allow easy closed eyes tacking when doing the first tacks holding things in place with one hand, you can do it with the others, but not as easily.
Believe it or not Stick welding is a good option provided you know what you are doing, if you don't have much practice you will hate using it on thin tubing though...
Luke.