How do you find the domain and range without having to graph? Like, is their an algebraic method? For example if I am asked to find the domain of $g(t) = \sqrt{t^2 + 6t}$ , how do I determine the range of this? 
Is their a universal algebraic method that I don't know about? 

Assuming that you are looking at a real function of a real variable you can determine the allowed domain as those values of t that produce a real result for g. In this case you need $t^2 + 6t \ge 0$ otherwise you are trying to get the square root of a negative number. Factorising $t(t+6) \ge 0$ with solutions $t \ge 0$ and $t \le -6$. So the allowed domain is $t \le -6$ and $t \ge 0$. The corresponding range is the values that g ranges over given this domain which can be seen to start from 0 (if $t = 0$ or $t=-6)$ and extend to $+\infty$
This changes if you allow g to be a complex function of a real variable, or a complex function of a complex variable.
The domain of a function is also often specified as a subset of the allowed domain, so you might have a function like g restriced by definition to a range $t \ge 0$.
