The textbook told me if $(\pi,E,M)$ is a vector bundle, if we have a local frame $s_i$ of $E$ over $U$ then forall vector $v_p$ at $p$ we have
$$v_p=c_1s_1+\cdots+c_ks_k$$
Then we define k maps $c_i$ from $\pi^{-1}(U)$ to $\mathrm{R}$ , so we can define a local trivialization of $E$ over $U$ by setting：
$$\Omega_U(p,v_p)=(p,c_1,…c_k)$$
But how can I prove that each $c_i$ is smooth?
The question is local on $M$, so let us assume that $E=U\times\mathbb{R}^k$ actually is trivial over $U$. We can think of each $s_i$ as a smooth function $U\to\mathbb{R}^k$, such that for each $p\in U$, the vectors $s_1(p),\dots,s_k(p)$ are a basis of $\mathbb{R}^k$.
Now let $A(p)$ be the $k\times k$ matrix whose columns are the vectors $s_i(p)$. The assumption that these vectors form a basis means that $A(p)$ is invertible. Moreover, your map $v_p\mapsto (c_1,c_2,\dots,c_k)$ is just the linear map $\mathbb{R}^k\to\mathbb{R}^k$ given by the inverse matrix $A(p)^{-1}$.
So to prove your map is smooth, you just need to know the matrix entries of $A(p)^{-1}$ are smooth functions of $p$. But this is true because the matrix entries of $A(p)$ are smooth functions of $p$, and the map taking an invertible matrix to its inverse is smooth (this follows from the explicit formula in terms of determinants).