How to specify algebraic relations between objects?

There are different ways to go about doing this. I'll set UpValues on e for the multiplication of the basis vectors as you've defined them, and I'll use replacement Rules for the simplification of products of vectors expanded in the basis.

First of all

Unprotect[e]
ClearAll@e
e /: NonCommutativeMultiply[e[i_], e[j_]] /; i < j := -NonCommutativeMultiply[e[j], e[i]]
e /: NonCommutativeMultiply[e[i_], e[i_]] := 1
Protect[e]

This sets up a "normal order" for the products of basis vectors. It might not be what you want, but it aids in simplifying compound expressions. Once you've run the lines above, then (non-commutative) products of the es are automatically sorted according to descending order left to right:

e[3] ** e[3]
(* 1 *)

and

e[1] ** e[2]
(* -e[2] ** e[1] *)

and

e[2] ** e[1]
(* e[2] ** e[1] *)

To extend this basis to a linear space, we form all formal linear combinations of these basis elements. Particular examples are

ClearAll[a,b]
vec1 = Sum[e[j] a[j], {j, 1, 3}]
vec2 = Sum[e[j] b[j], {j, 1, 3}]
(* a[1] e[1] + a[2] e[2] + a[3] e[3] *)
(* b[1] e[1] + b[2] e[2] + b[3] e[3] *)

Then, we can implement some linearity rules and scalar rules

ncmRules = {
   a___ ** (-b_) ** c___ :> -a ** b ** c,
   a_Plus ** b_ :> (# ** b & /@ a),
   a_ ** b_Plus :> (a ** # & /@ b),
   x___ ** a_ ** y_ /; FreeQ[a, e] :> a x ** y,
   x_ ** a_ ** y___ /; FreeQ[a, e] :> a x ** y,
   x___ ** (a_ y_) ** z_ /; FreeQ[a, e] :> a x ** y ** z,
   (x_) ** (a_ y_) ** z___ /; FreeQ[a, e] :> a x ** y ** z,
   NonCommutativeMultiply[a_] :> a
 };

and define a function that applies these rules:

ncmSimplify[expr_] := expr //. ncmRules

For instance,

Collect[vec1 ** vec2 // ncmSimplify, _e ** _e]
(* a[1] b[1] + a[2] b[2] + a[3] b[3]
    + (a[2] b[1] - a[1] b[2]) e[2] ** e[1]
    + (a[3] b[1] - a[1] b[3]) e[3] ** e[1]
    + (a[3] b[2] - a[2] b[3]) e[3] ** e[2] *)

and

e[1] ** e[2] ** e[1] ** e[2] // ncmSimplify
(* -1 *)

There is a new paclet for doing Geometric Algebra:

PacletInstall["https://wolfr.am/N9OenlOc"]
<< GeometricAlgebra`;

You can construct multivectors of any algebra with it, and use it in computations:

{e1, e2, e3} = MultivectorBasis[GeometricAlgebra[3], 1];

e1 ** e2 ** e1 ** e2
(e1 ** e2) ^ 2
e1 ^ (3/5)

And more:

Inverse[2 e1]
MultivectorFunction[Exp, e1]
(e1 + 3 e2)["Normalize"]
(e1 + 2 e2 + e3)["Conjugate"]
e1["Dual"]
MultivectorMatrix[e1 + I e2]
ConvertGeometricAlgebra[e1, GeometricAlgebra[0, 3]]