what is the fastest/cheapest way to patent an idea, how much/how long should it take?
There's no easy answer to that question.
You can apply for a "provisional patent" which will permit you to mark your invention as "patent pending", however it will not give you the same rights as an actual patent. It is not very expensive, though.
If you apply for a provisional patent, you should immediately work on submitting a formal patent application. If your entire invention (tfor which you submit the formal application) is disclosed properly within the provisional patent application, then if and when you are granted an actual utility patent, you may be able to have its effective date of your invention be set to the date when you submitted your provisional application.
There are some possible downsides to going through the provisional patent route, however. For example, I don't think that you can get a provisional patent for countries other than the United States, and in fact there may even be some issues with patentability in foreign countries if you have filed a provisional application in the United States first. Also, provisional patents expire in one year, and so if you have not filed a proper patent application during that year, it is possible that you could lose your ability to patent your invention at all. I don't recall offhand whether or not provisional patent applications are disclosed in the way that regular applications are. If they are disclosed, then it's possible that a foreign competitor could get the jump on you in filing for a foreign patent.
So if your invention has real commercial possibilities, it's a good idea to get a registered patent attorney involved up front. (Translation: $$$$)
The US patent office is currently swamped with a huge backlog of patent applications, and so under the best of circumstances it can take years to get a patent issued. Typically there will be a lot of back and forth activity with the USPTO that can stretch the process out even longer.
for info on provisional patents.
There is no such thing as a "cheap" patent. The companies that claim to apply for patents for you are 100% scams.
The actual fees for filing the porvisional and actual patents are on the US Patens Office website. Patent lawyer fees are the real cost. Patens can take years to get approved.