Relationship between groups and spaces From the definition on Wikipedia: In mathematics, a space is a set (sometimes called a universe) with some added structure.
 A group seems to satisfy this definition. For example, we know that Lie groups are differentiable manifolds. 
So my question is: is it proper to say that a group is also a kind of spaces, just like vector spaces and topological spaces, but with different or simpler structure? 
 A: Sure, it can be useful to attempt to think about mathematical concepts in a way that goes against the grain a little bit. Here are a few ways of thinking of groups as spaces. I suppose it should go without saying that these are very useful ways of thinking about groups.
There are topological groups, which you could define as topological spaces that admit the structure of a group, or  as groups where the defining features (multiplication, inversion) are continuous maps. As you mention, Lie groups  are a good example of topological groups. Of course, any group may be given the discrete topology, but even  this is relevant, for instance, in some definitions of a properly discontinuous group action.
Another topology that any group can be given is the profinite topology, where a neighborhood basis  of the identity in $G$ is the set of finite index subgroups of $G$. Useful properties of infinite groups, like residual finiteness, are topological statements about the profinite topology.
If $S$ is a generating set for a group $G$, $G$ can be given the word metric, $d_S$, where $d_S(g,h)$ is the  minimum number of elements of $S$ required to write  $gh^{-1}$ in terms  of this basis. Although this gives every group the discrete topology, when $S$ is a finite set, one can learn quite a lot about $G$ from $d_S$,  at least as distances go to infinity! (This is a starting  point for geometric group theory.)
