A lot will depend on how much fuel you personally use. Then the next is how the fuels in question are available in what quantities where you will be.

A few notes from my personal experience:

1. I have done a good bit of touring with pop can stoves using alcohol. Where I have toured I have been able to find 12 ounce bottles of Yellow Heet. I find it works out well enough and my alcohol cooking kit is the lightest i have found. If there is a need to carry large amounts of fuel for some reason the weight advantage may shift to some more btu dense fuel, but I doubt I'll ever tour somewhere I'll restock that infrequently. Fire bans can be an issue though. They often ban stoves with no off switch/valve.
2. Canister stoves like the pocket rocket or similar are a little heavier. The simmer a little better and boil a little quicker. I tend to use one when I am taking a stove for a small group or want an off switch. I have had issues finding canisters even at a time when every walmart was supposed to sell them according to posts on the forums. That was quite a few years ago though.
3. I have a couple liquid fuel stoves that I'd consider using on a tour. One officially burns only white gas and the other is a multi fuel model. In practice either would probably burn gasoline in a pinch, but I'd prefer to only burn it in the multifuel one and that maybe only in a pinch. White gas (coleman fuel) comes in larger size containers than I prefer to carry, but I did find a workaround other than gasoline. It turns out that zippo or ronson sell lighter fluid for cigarette lighters and hand warmers that is the same thing as coleman fuel. It comes in 4, 6, 8, and 12 ounce bottles and is pretty widely available. I found it locally in a few places here in my home town and a google search seems to show it as pretty widely available. It was in 4 and 8 ounce sizes for about $2 here in my home town. Watch out for the charcoal lighter fluid though. That is usually closer to kerosene. It will burn in my multifuel stove but it burns poorly unless I swap to a special kerosene jet. Also it comes in larger sizes than what I want to carry at a time.

FWIW, my stoves I mentioned in item 4 are a SVEA 123 and an MSR International.

Edited to add that I have tested the zippo/ronson stuff and used it for a short trip, but not a bike tour. Also you may or may not be able to mail it to yourself via general delivery surface mail only. It would need to be labeled ORM-D. I tend to think it is okay for limited quantities based on the research I did when I wanted to mail isobutane canisters. I also did see some references on line to folks doing it. It definitely is shipped from Amazon and others via fedex and ups. I suppose you might be able to mail regular coleman fuel the same way, but the amount is very limited and I'd avoid repackaging it. A factory sealed small quantity would be most likely to be withing the acceptable realm of the USPS. I have no idea about the rules elsewhere.

The ORM-D regulations for flamible stuff are often disputed when brought up, but I decided it was okay to mail the canisters when I looked into it. There was a size limitation, but I don't recall what the deal was. I only wanted to mail small canisters so It wasn't something that concerned me much.