The essence of what you're getting at is the equivalence of these definitions:
A spanning tree of a connected graph $G$ is subtree $T$ of $G$ for which every vertex of $G$ is incident to an edge of $T$.
A spanning tree of a connected graph $G$ is a maximal set of edges containing no cycles.
Actually there is a third equivalent definition, sort of combining the two ideas above:
A spanning tree of a connected graph $G$ is a minimal set of edges containing all vertices.
Once you understand these equivalences (the proofs are better discovered than read), it will become clear how to show every graph has a spanning tree in both ways: building up using Def. 2, and building down using Def. 1. Indeed in either case the proof will jump out at you!
If you're really stuck here's an outline of the equivalence of the two definitions: suppose you have a spanning tree in the second sense. Could it be disconnected? (No. Why?) Could it be missing a vertex? (No. Why? You could add an edge.)
Suppose you have a spanning tree in the first sense, clearly it has no cycles, but could you add any edge to it? (No. Why? You'd create a cycle.)