Estimating the complexity of an algorithm by looking at code Time complexity seems rather a difficult notion for people who were not professionally trained in math or computer science. I understand it informally and can high-level compare which one is better and which one is worse (say: polynomial time algorithm is more efficient/faster than exponential time algorithm), but my question is how can one roughly assess the time complexity of his/her own algorithm? 

A good example is the grand-canonical problem of integer factorization. Say one tries to write a new method to factor these awfully big integers. We know that the input has x-decimal digits and b-bytes. What does technically make an algorithm inefficient from the start? (say nested loops kills the program already and it's not even worth in-depth measurement? Does one need to obey certain programming rigours in order to ensure time complexity is capped at certain (reasonable) level)? 

Sorry if this sounds really informal and amateurish, but I am just very curious about this kind of things, though have little professional knowledge to understand it on my own.
This question is rather informal, and you may enjoy reading this.
What you are asking about is a topic discussed often in Computer Science and Mathematics (perhaps all subjects, but my perspective is not so broad) - "developing an intuition". Often, students are looking for some sort of cheat code to solving problems quickly without going through all of the steps. The honest answer is that you can only develop such intuition and speed by doing precisely the opposite - practicing slowly and without any sort of shortcut. 

Practice calculating time complexity - it really isn't too difficult. A quick google search gave me this page, which is pretty informal and may suit the needs of some practice problems. Once you have gone through the process enough times, you will ascend to a sought-after plane of intuition.

Having very little professional knowledge is not an issue. If the example page I sent is too formal, there are plenty of other pages on the web explaining time complexity. You just have to be willing to sit down and put some hours into it (which no different than what a "professional" does to get the same knowledge, except perhaps they pay more for it?).