In classical mechanics, an observable is a bounded continuous function $f$ on a phase space, which can be assumed to be $\mathbb{R}^{n}$ for some $n$. If $f$ is differentiable, then: $$\frac{df}{dt}= \{f,H\}$$ where $H$ is the Hamiltonian of the theory and $\{\cdot,\cdot\}$ denotes the usual Poisson brackets.
It is said that the above expression implies that $H$ generates the action of time translations. I have no background in Lie groups and Lie algebras, and I would like to understand sentences such as the latter. In short, I would be able to understand what is a Lie group and a Lie algebra, what is an action of time translation, how it is connected with infinitesimal generators, symmetries in physics and so on. For this reason, I am looking for good mathematical references.
I am really interested in its applications to physics, specially classical mechanics, but I am looking for mathematical (i.e. rigorous) resources. In addition, I know these concepts arise naturally in differential topology, but I am not very familiar with manifolds and so on, so I would like a discussion which avoids these concepts. Any suggestion is very welcome!