I'm confused with the relation between these two. I'll summarize what I know so far and explain my questions.
So for a principal $G$-bundle $P$, we can define a principal $G$-connection on it, we can then have an induced connection on the associated fiber bundle $F$. Also, this connection is a 1-form with values in the Lie algebra $\mathfrak{g}$ of $G$.
On the other hand, we have for a vector bundle $V$ with structure group $G$, we can then have a subbundle of its frame bundle $F(V)$, which is a principal $G$-bundle. Also, this $G$-structure on the vector bundle is usually associated to some extra structure. For example, if $G=O(n)$, that means we have a fiber metric.
I wonder if the following statements are true:
For a connection on the $V$, can we induce from it a principal $G$-connection on the $F(V)$? If this is not always the case, can we put a little extra structure on the connection that could make this true? e.g. to make the connection compactible with the $G$-structure on the vector bundle. (But I only know how to describe this compactibility for groups like $O(n),U(n)$)
For the 1-form with values in $g$ that represents the principal $G$-connection on $P$, what does the induced connection on $V$ look like? Since a connection on a vector bundle is always represented by a matrix of 1-form, do we always have the 1-form matrix takes value in $\mathfrak{g}$ (i.e. that is to say connection matrix itself as a matrix is in $\mathfrak{g}$)
P.S. Any help is a ppreciated. Also, I did not find some good reference about this topic, so please let me know if you know some good places that can answer these questions.
So the answer to (1.) is positive. If you have a connection on a vector bundle you also obtain a connection form on the associated frame bundle.
In general you need to work a bit to see this. If $\mathcal{A}$ is a connection form on a principal $G$-bundle $P\to M$, then for each local section $s:U\to P$, $U\subset M$ in $P$ you can locally represent it as $s^*\mathcal{A}\in\Omega^1(U,\mathfrak{g})$. By chosing a family of sections whose domains cover $M$ you have now stored all information on $\mathcal{A}$ in these local forms and can reconstruct $\mathcal{A}$ from these. Moreover, they are subdued to a certain transformation formula if you change the section, $$ s_i^*\mathcal{A}=\text{Ad}_{g_{ij}^{-1}}\circ s_j^*\mathcal{A}+g_{ij}^*\,\mu^{\text{MC}}, $$ with the Maurer-Cartan form on $G$ and the functions $g_{ij}:U_i\cap U_j\to G$ which translate the sections $s_i:U_i\to P$ and $s_j:U_j\to P$ into one another on their common domain.
Now do the converse, whenever you have a family of $\mathfrak{g}$-valued 1-forms $\{\mathcal{A}_i\in\Omega(U_i;\mathfrak{g})\}_i$, whose domains $U_i$ cover the basis $M$ and a family of local sections $\{s_i:U_i\to P\}_i$ such that the above formula holds, then you can built a connection form $\mathcal{A}$ on all of $P$ such that $s_i^*\mathcal{A}=\mathcal{A}_i$.
If you now start with a vector bundle $V\to M$ with fiber type $V_0$ and a connection $\nabla$ thereon, you can choose a local frame $e_1,\dots,e_n:U\to V$ and represent the action of $\nabla$ as a matrix of 1-forms, $\nabla e_i=\sum_j \omega_{ji}\otimes s_j$. That way you can write $\omega$ as a 1-form with values in $\text{End}(V)$. The endomorphism bundle $\text{End}(V)$ is now of fiber type $\text{End}(V_0)$, which is exactly the Lie algebra of the frame bundle's structure group $\text{GL}(V_0)$. If you had additional structure, e.g. a fiber metric, you would have chosen an orthonormal frame, would have received a skew symmetric matrix which in in the Lie algebra of $\text{O}(n)$, and so on.
For these Lie-algebra valued forms you can now show the following: The chosen frame for finding the form $\omega$ defines a section in the frame bundle. If you now cover your whole basis $M\subset\bigcup_iU_i$ with domains of local frames of $V$, or sections of its frame bundle, respectively, then you find that the above forms fulfill the required transformation law from above, and thus you can build them together in a connection form on the frame bundle (or subbundles with a smaller structure group in the same way).
For (2.) you also need to work a bit, and I will not work that out in detail. But, say you have a principal $G$-bundle $P\to M$ and an associated vector bundle $V:=P\times_{\rho,G} V_0\to M$ via a representation $\rho$ of $G$ on $V_0$, then the crucial ingredient is an isomorphism between $\Omega_\text{hor}(P;V_0)^{(G,\rho)}$ and $\Omega(M,V)$. Hereby, $\Omega_\text{hor}(P;V_0)^{(G,\rho)}$ is the space of differential forms on $P$ with values in $V_0$, which transform along the fibers of $P$ via the representation $\rho$, and which are horizontal.
By means of this isomorphism you can push back and forth any action on commensurable $V_0$-values forms on $P$ to $V$-valued forms on $M$, particularly you can push the horizontal derivative on horizontal forms to $V$-valued forms on $M$ and easily show that this action defines a covariant derivative on $V$ (sections in $V$ are just $V$-valued 0-forms). This association is again "inverse" to the above in the sense that, if you have started with $V$ and $\nabla$ thereon as above, and let $\mathcal{A}^\nabla$ be the associated connection form on the frame bundle of $V$, then what I just described reproduces $\nabla$ on $V\cong P\times_{(G;\text{id})}V_0$ (with the trivial representation $\text{id}:\text{GL}(V_0)\to\text{GL}(V_0)$).
But on one thing you need to watch out a bit. While a connection form on a principal bundle is a Lie algebra valued form in general (by definition), a connection on a vector bundle is not, only locally in a frame as I described above. So I think keeping in mind that "connections on vector bundles are Lie algebra valued forms" is a bit misleading, since if they were globally such forms, that would kind of kill their property as a way of covariantly differentiating.
I hope I could help you a bit, though the second part might be a bit large or general of a question for MSE (you can white whole math book chapters on this topic).
Since you asked, I liked the book of Helga Baum. At some points it may be hard to follow, but I think she develops a good viewpoint on all these things. I think it's only available in German, but I also found various lecture notes in English which are based on this book on google (but forgot who the authors were).