Nowadays, the #1 method for predicting the next number from a sequence (assuming the sequence has come up in a "natural" way) is to look it up in the Online Encyclopedia of Integer Sequences. In his 1973 book, A Handbook of Integer Sequences, Sloane gives some suggestions as to what to do if your sequence is not in the Encyclopedia/Handbook. These include,
Add or subtract 1 or 2 from all the terms, and try looking it up again;
Multiply all the terms by 2, or divide by any common factor, and try looking it up again;
Look for a recurrence.
Sloane elaborates on this last suggestion. He mentions the method of differences, where you replace the sequence $a_0,a_1,\dots$ with $a_1-a_0,a_2-a_1,\dots$ and, if necessary, repeat the differencing, until you get something with an obvious pattern. Of course, then you have to know what to do with a recurrence once you have one, but that's another story.
Sloane also says that if a sequence is close to a known sequence, you can try subtracting off the known sequence, and then dealing with the residual by one of the above methods.
If the ratios $a_{n+1}/a_n$ seem to be close to a recognizable sequence $r_n$, then look at the sequence given by $a_{n+1}-r_na_n$.
Factoring the numbers in a sequence, or in a sequence close to the given sequence, will often give a clue as to what is going on.
For examples of all these principles (and others that I haven't mentioned) in operation, I refer you to the Handbook.
Best Answer
Look at the differences between consecutive terms:
$$\begin{align*} &35-12=23\\ &81-35=46\\ &173-81=92\\ &357-173=184 \end{align*}$$
Can you find a pattern now?