Is a sequence (123... abc... qwerty and so on) easier to be broken? Are there attempts made by computers that consider these possibilities first? I have no idea how to crack a password, but I'd like to understand it, so that I can minimize the chance, whereas keeping my passwords as simple as possible.
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Yes, good password crackers will use lists of well known passwords and test against them first. Then dictionary attacks and variations on them. Finally they will set about to do letter and special character combinations. So yes, the 'easy' passwords get broken extremely fast.
Consider moving to a passphrase, which is longer but meaningful. Lyrics to a song or your favorite line from a show, with numbers switched for letters as an example.