This is basically a comment, but got too long. All the algebraic groups mentioned in the question are "simple" (also called "quasi-simple") over an algebraically closed field in the sense of having no closed connected normal subgroups except the entire group and the trivial group. Groups of type $G_2$ are in fact simple as abstract groups, but others might have a nontrivial finite center. For reductive groups there can be a torus in the center, as happens with $GL_n$; but here the derived group $SL_n$ is simply connected. On the other hand, $SO_n$ fails to be simply connected but is its own derived group.
This kind of group structure for classical groups has been exposed in many books, though for the exceptional groups it's necessary to take a more unified viewpoint.

Concerning references, the basic structure and classification theory (going back to the 1956-58 Chevalley seminar) is worked out over an algebraically closed field in a number of textbooks with the same title *Linear Algebraic Groups* by Borel, Springer, and myself. Classifying "forms" over smaller fields is dealt with only partway in these books, while the scheme approach is found in SGA3, Demazure-Gabriel, and more recently the monograph *Pseudo-reductive Groups* by Conrad, Gabber, Prasad. In any case, it's important when working in this algebraic setting over fields which might not be of characteristic 0 to clarify the notion of "simply connected". Chevalley's notion, which agrees for the related complex Lie groups with the topological notion, depends just on a (split) maximal torus and its character group relative to the given reductive group.