OP, you're going to find as many definitions of 'sign' as you'll find semiologists, semioticians, semeioticians, semeiologists, and students of those semioticians, semeiologists, semeioticians, and semiologists.
The big two that we're bouncing between right now are Saussure and Peirce. The relevant (though by no means only) texts are Saussure's "Course in General Linguistics" and Peirce's "What is a Sign?".
Saussure
A 'sign,' sometimes poorly translated as 'signal,' is really the "marriage" of the signifier and the signified. So when we say 'sign,' what we mean (if we're Saussurian, that is) is 'signifier-signified.' The next step, then, is to define these two similar-looking words, and this is where things often get confusing.
What Saussure means by 'signifier' is the 'sound-image' of the sign. Its phonetics, its sequence of consonants and vowels. Saussure will also treat the written-form of the sound-image as a signifier, too, but it is important to remember that he begins his analysis with the 'sound-image.'
'Signified' is the one that complicates things. Really, Saussure means: concept. NOT the actual thing or object. He is very clear on this: our analysis of a sign is an analysis of langue, which we often translate in English as 'language-system.' We are not analyzing parole, or 'speech.' To say that the signified is the object or thing is to import a metaphysical claim into the analysis. It leads us down the path of inappropriately giving positive definitions to signs, when Saussure insists that really, all signs are defined negatively. This analysis is purely of the system of language.
A sign, in langue, means what it is not. The meaning of the sign is determined negatively by the meanings of every other element in the language-system (langue). Although it sounds silly, there is a very good reason for why this is the case: it allows for the system of language to remain internally coherent and complete. For example, imagine you have a langue with a sign for the colour red, but no sign for the colour pink. In this case, every colour that the English langue would claim to be 'pink' would simply be rendered as 'red,' because there is no sign for 'pink' in our hypothetical langue. Now imagine that a sign for pink has been invented in our hypothetical langue. In order for this sign to have its meaning, it will first require a shrinking of the scope for the existing sign for 'red.' 'Red' will yield some of its scope in order for 'pink' to have its own meaning. Thus, 'pink' can be understood as meaning 'not-red.' This is a simplified case: really, every sign in the system is in this kind of close relationship.
Saussure insists that the signs of langue (which, recall, is only one way of looking at langage, or, language; the other way being parole, or, speech) are constituted by an arbitrary connection between the signifier and the signified. There is no good reason why "fromage" should signify the concept of coagulated milk proteins. We know this because "cheese" is another signifier for that same (really, not the same, since any two language-systems are, by necessity, incommensurable; but just go with me here) signified.
However, we should not take this as gospel. Benveniste and others have shown that, actually, the connection between the signifier and the signified is often understood - in fact, needs to be - as necessary. I won't pursue this right now, because it is not within the scope of this analysis, and it involves a lot of tricky nuances between langue and parole. I only mention it in order to remind us that Saussure is the beginning of this particular branch of semiotics; he is certainly not the only voice.
As for what Saussure thinks of symbols? /u/AnxiousReginald has already told us - symbols are motivated connections between "signals" (better translated as "signifier") and "significations" (better translated as "signified"). Saussure suggests that, really, symbols appear as a kind of special case of "signifier."
Peirce
/u/mentalmobius has done a great job breaking down the three types of, or ways of being, signs.
One thing we cannot do here is try to map it into Saussure's framework, or vice versa. They are not compatible. We can analogize, and consider that each of Peirce's signs are special configurations of the signifier-signified dyad, but really, this endeavour is futile. Saussure is a dichotomic thinker. Langage splits into langue and parole. Sign splits into signifier and signified. Our analysis can either be synchronic or diachronic.
Peirce is trichotomic. Everything comes in threes. We would need to flatten one of his dimensions in order to compress him in Saussurean terms. Likewise, we would need to extrapolate a third dimension in order to disperse Saussure into Peircean terms.
Signs are much less limited for Peirce. We think in signs. We, ourselves, are signs. The world we interact with is a world of signs. In fact, you could say that our experience is merely endless semeiosis, or sign-proliferation, that can be analyzed either in terms of likenesses, indications, or symbolizations.
Saussure has two kinds of analysis, one that examines a snapshot of the langue, and another that examines the development of that langue from one point in time to another. He has no explanation (and he doesn't need one) for how this change occurs. Peirce builds this in to his semeiotics: Symbols grow. Signs are subject to development, or evolution. A relevant passage from Peirce's "What is a Sign?" (section 8):
Symbols grow. They come into being by development out of other signs, particularly from likenesses or from mixed signs partaking of the nature of likenesses and symbols. We think only in signs. These mental signs are of mixed nature; the symbol-parts of them are called concepts. If a man makes a new symbol, it is by thoughts involving concepts. So it is only out of symbols that a new symbol can grow. Omne symbolum de symbolo. A symbol, once in being, spreads among the peoples. In use and in experience, its meaning grows. Such words as force, law, wealth, marriage, bear for us very different meanings from those they bore to our barbarous ancestors. The symbol may, with Emerson’s sphynx, say to man,
Of thine eye I am eyebeam.
Symbols, as the third and therefore broadest class of signs (insofar as, for Peirce, the third will always be the mediation between the first and the second), provide language, thought, humanity, the world, etc, with its generative impetus.