That does not answer really the question, but I don't think that computer algebra is really about solving equations. For most kind of equations I can think about (polynomial equations, ordinary differential equations, etc), a closed-form solution using predefined primitives usually does not exist, and when it does it is less useful than the equation itself. For example, consider univariate polynomial equations. It is not always possible to solve these equations using nth-roots, and when it is, it is smarter to represent a solution of such an equation by the equation itself. With this representation you can perform addition and multiplication easily.
I my opinion, computer algebra is more about manipulating the equations themselves.
For example, you have two univariate polynomial equations, virtually defining two algebraic numbers, and you want to compute the equation satisfied by the sum of these two numbers. Most of time you want the equality to be decidable.
You have to delimit carefully the class of objects you are working with because if the class is too wide the equality decision problem may quickly become unsolvable, see Richardson's theorem
Of course there is numerous situations in computer algebra where one want to solve an equation.
Of fundamental importance, there are linear equations, they are ubiquitous.
I think also of polynomial system. In the domain of ordinary differential equations, I think of rational resolution: given an ODE, you want to find all rational solutions; or power series resolution.
To answer more precisely your question, I think you should implement first the resolution of linear system of equations. You may use Gauss' elimination. When efficiency will become an issue, you may have a look at Bareiss' algorithm. After that there is a whole bunch of various and specialised algorithms to consider.
After that, all depends on what you are interested in. You could implement algebraic numbers, polynomial factorization. You might also be interested in numerical computation, fancy linear algebra, arbitrary precision computation, etc.