Is it correct to say: "A, B, then C" instead of "A, B, and then C"?

I'd originally posted this as a comment. But because I disagree with both answers here, I'll post it as an answer so anyone who agrees with me can upvote it.

You can separate any or all of the steps using either or both of and and then. It's really just a matter of style with very little significance - except that if you use neither conjunction, even before the last step, it can sound a bit brusque.

I'll also add that personally I think it's stylistically odd to reduce the conjunctive forms as you list the steps. For example, I don't much like...

*Do step1, and then step2, then step3.

...it's more common to increase them...

Do step1, step2, then step3 (or ...step2, and step3).

Do step1, then step2, and then step3.

Yes, you can coordinate all three without and. This is called asyndetic coordination, which means that there's no overt coordinator present:

Syndetic coordination: A, B, and C
Asyndetic coordination: A, B, C

In your example, the third conjunct contains the adverb then functioning as a connective adjunct:

[ Go to google.com ]_A, [ enter miserable failure ]_B, [ then click I'm Feeling Lucky ]_C.

You could insert the overt coordinator and before C, or before both B and C. But you don't have to.

When there's no overt coordinator, commas are generally required. In the following example, I've omitted what would be the Oxford comma, making the sentence unacceptable:

*Go to google.com, enter miserable failure then click I'm Feeling Lucky.

Compare this to one of the most famous examples of asyndetic coordination:

I came, I saw, I conquered. 　 　 (contains obligatory comma)
*I came, I saw I conquered. 　　 (missing obligatory comma)

If then truly were a coordinator, we'd be able to join three items without an Oxford comma, or pair two items without any commas:

I ate food, then I went to the store. 　 　 (contains obligatory comma)
*I ate food then I went to the store. 　　 (missing obligatory comma)

But we can't, so it's not.

Of course, this sort of coordination isn't always appropriate. In particular, when the conjuncts seem only loosely related (or not related at all!), people are likely to notice how they're put together and call the resulting coordination a "comma splice". To some people, the difference between asyndeton and a comma splice is whether or not they like the result:

I came, I saw, I conquered.　　　　　　　　(I like it, must be asyndeton.)
^??Computer games are fun, I'm hungry.　　(I don't like it, must be a comma splice!)

So use your own judgment. If you're joining clauses that are only loosely related and you think they need an overt coordinator, put one in. (Or don't coordinate them.)

This reference puts it best:

This is a comma splice, a faulty sentence construction in which a comma tries to hold together two independent clauses all by itself: the comma needs a coordinating conjunction to help out, and the word then simply doesn't work that way.

A practical piece of advice for remembering coordinating conjunctions from the same site:

It may help you remember coordinating conjunctions by recalling that they all have fewer than four letters. Also, remember the acronym FANBOYS: For-And-Nor-But-Or-Yet-So. Be careful of the words then and now; neither is a coordinating conjunction, so what we say about coordinating conjunctions' roles in a sentence and punctuation does not apply to those two words.

So, to answer the question -- no, the sentence needs to read:

Go to google.com, enter miserable failure, and then click I'm Feeling Lucky.

As far as I know, "then" is not a conjunction here, and it cannot be used alone without "and".

Go to google.com, enter miserable failure, and then click I'm Feeling Lucky.

If you don't want to use "and", then you can change the punctuation:

Go to google.com. Enter miserable failure. Then click I'm Feeling Lucky.