The Wikipedia references are a reasonable starting point in the sense that some of them contain Voevodsky's construction(s) and related expositions. Beyond that the literature on motives is enormous and difficult and relies on many ideas that are not apparent from the papers themselves -- not to mention unpublished letters and oral tradition found in a few major mathematical centers. The state of documentation is far better in the internet era but this is still a highly technical field that is difficult to penetrate by casual reading. In one respect the main breakthrough made by Voevodsky is that he created a theory with a framework that can be learned (e.g., by topologists) knowing scheme theory and a bounded number of other things, whereas before that motives were considered a pinnacle of inaccessible mystery, or a foundation-less heuristic used informally.
As for what is "suitable", Voevodsky proved the universality of motivic cohomology as defined in his approach; in the 2-category of homology theories satisfying some axioms, he constructs an initial object (similar to characterizing singular homology as the universal model of Eilenberg-Steenrod axioms). Also, he builds a motivic homotopy theory with analogies to the topological theory, and not "only"(!!) the motivic homology and cohomology. There are other respects in which his approach is better motivated (so to speak) or conceptual compared to other approaches. (Grothendieck's approach is also conceptual but assumes Standard Conjectures and the work on mixed motives was not published.) So although nobody knows the ultimate construction of mixed motives, Voevodsky's approach is the leading contender at present.