It is certainly possible to give the definition of a quotient of a variety by an algebraic group without mentioning algebraic stacks or spaces.

There are two distinctly different situations here, depending on whether the group is finite (and discrete) or a general algebraic group.

In the finite case things are a lot easier. Then you could take as your definition of a quotient that on an affine scheme given as the spectrum of a ring R, the quotient is the spectrum of the ring of invariants $R^G$. (This is not a good definition, but you could.) For general schemes you can try to glue together open affines to define a quotient globally, but this will not always produce a scheme (roughly speaking, you might have to identify too many points). When the original scheme is quasiprojective, this gluing procedure will give you a scheme (folklore), and in general it will always produce an algebraic space (Deligne).

In the more general case of algebraic groups one needs to be more careful and start distinguishing between different definitions of quotients, and start keeping track of whether your group is geometrically reductive or linearly reductive or neither. The quotient definition given by James Borger is called the categorical quotient. It is unfortunately not well behaved in the category of schemes. (Example: let $\mathbb{C}^\ast$ act on $\mathbb{C}^2$ by scaling. Then a categorical quotient exists in the category of schemes (!), however it is a single point.) The basic problem is that most properties that one wants from a quotient, like "the preimage of a point in X/G is a single G-orbit in X" will not hold unless one imposes extra conditions on top of the categorical one.

More well behaved notions are in particular good quotient and geometric quotient, but beware that there is here a morass of different definitions of quotients which can be more or less nice (there are things called weak quotients and semi-geometric quotients and probably more that I don't know, and these can be modified by adjectives like "uniform" or "universal" or "categorical", and there are nontrivial combinations of these properties like "good geometric" so these notions of quotients are not linearly ordered). There are also some differences in the literature between how these properties are defined. The point that makes finite groups so much easier is that for a finite group all these types of quotient will coincide anyway.

Just as an example let me give the definition of a geometric quotient $\pi : X \to X/G$:

1. $\pi$ is G-equivariant;
2. the geometric fibers of $\pi$ are the orbits of geometric points on $X$ (i.e. the property I mentioned earlier, but it is required only for points with coordinates in an algebraically closed field);
3. $U \subset X/G$ is open if and only if $\pi^{-1}(U)$ is open;
4. $(\pi_\ast O_X)^G = O_{X/G}$.

(Quoted from GIT)

The most important work in this area is the slightly intimidating GIT (Geometric Invariant Theory) by David Mumford. The main construction of the book shows that when the group is reductive you can find open dense subsets traditionally denotes $X^s$ and $X^{ss}$ such that the $X^s$ has a geometric quotient and $X^{ss}$ has a good quotient which is projective. It is written like a textbook so in principle you can start reading on page one, but there are friendlier introductions out there. I like Dolgachev's Lectures on Invariant Theory.

It is certainly possible to give the definition of a quotient of a variety by an algebraic group without mentioning algebraic stacks or spaces.

There are two distinctly different situations here, depending on whether the group is finite (and discrete) or a general algebraic group.

In the finite case things are a lot easier. Then you could take as your definition of a quotient that on an affine scheme given as the spectrum of a ring R, the quotient is the spectrum of the ring of invariants $R^G$. (This is not a good definition, but you could.) For general schemes you can try to glue together open affines to define a quotient globally, but this will not always produce a scheme (roughly speaking, you might have to identify too many points). When the original scheme is quasiprojective, this gluing procedure will give you a scheme (folklore), and in general it will always produce an algebraic space (Deligne).

In the more general case of algebraic groups one needs to be more careful and start distinguishing between different definitions of quotients, and start keeping track of whether your group is geometrically reductive or linearly reductive or neither. The quotient definition given by James Borger is called the categorical quotient. It is unfortunately not well behaved in the category of schemes. (Example: let $\mathbb{C}^\ast$ act on $\mathbb{C}^2$ by scaling. Then a categorical quotient exists in the category of schemes (!), however it is a single point.) The basic problem is that most properties that one wants from a quotient, like "the preimage of a point in X/G is a single G-orbit in X" will not hold unless one imposes extra conditions on top of the categorical one.

More well behaved notions are in particular good quotient and geometric quotient, but beware that there is here a morass of different definitions of quotients which can be more or less nice (there are things called weak quotients and semi-geometric quotients and probably more that I don't know, and these can be modified by adjectives like "uniform" or "universal" or "categorical", and there are nontrivial combinations of these properties like "good geometric" so these notions of quotients are not linearly ordered). There are also some differences in the literature between how these properties are defined. The point that makes finite groups so much easier is that for a finite group all these types of quotient will coincide anyway.

Just as an example let me give the definition of a geometric quotient $\pi : X \to X/G$:

1. $\pi$ is G-equivariant;
2. the geometric fibers of $\pi$ are the orbits of geometric points on $X$ (i.e. the property I mentioned earlier, but it is required only for points with coordinates in an algebraically closed field);
3. $U \subset X/G$ is open if and only if $\pi^{-1}(U)$ is open;
4. $(\pi_\ast O_X)^G = O_{X/G}$.

(Quoted from GIT)

The most important work in this area is the slightly intimidating GIT (Geometric Invariant Theory) by David Mumford. The main construction of the book shows that when the group is reductive and acts linearly on a projective variety $X$, there are canonically defined open dense subsets traditionally denotes $X^s$ and $X^{ss}$ such that the $X^s$ has a geometric quotient and $X^{ss}$ has a good quotient which is projective. When X is not projective one has to work with so called L-linear actions where L is a line bundle on X. This reduces to the case X projective by taking $L = O(1)$. GIT is written like a textbook so in principle you can start reading on page one, but there are friendlier introductions out there. I like Dolgachev's Lectures on Invariant Theory.