Show that if $f_1$ and $f_2$ are integrable on $[a, b]$ then $c_1f_1 + c_2f_2$ is integrable on $[a, b]$ and $$\int_a^b[c_1f_1(x) + c_2f_2(x)]dx = c_1\int_a^bf_1(x)dx + c_2\int_a^bf_2(x)dx$$
For the first part I think using the definition: For every $\epsilon > 0$ there exists a partition $P \in \mathcal{P}_{[a, b]}$ such that $U_f(P) - L_f(P) < \epsilon$ would be the most convenient method.
But, I don't really know how to make this definition useful here. Any help would be appreciated.
Let's start by proving $\int k f(x) dx = k\int f(x) dx$
Suppose we have $F(x)$ so that $F'(x) = f(x)$
We know that $\forall k \in \mathbb{R}, (kF(x))' = kF'(x) = kf(x)$
So $kF(x)$ is an anti-derivative of $kf(x)$
Hence :
$$\int kf(x)dx = kF(x)+c = k\int f(x) dx$$
In a similar fashion we can prove that $\int f+g = \int f + \int g$ :
Let's have $F'(x) = f(x)$ and $G'(x) = g(x)$
We know that $(F(x)+G(x))' = F'(x)+G'(x) = f(x)+g(x)$
So $F(x)+G(x)$ is an anti-derivative of $f(x)+g(x)$
Hence :
$$\int (f(g)+g(x)) dx = F(x)+G(x)+c = \int f(x)dx + \int g(x)dx$$
EDIT : To prove the linearity of integrability you need to look at Riemann integral definition.
If $f$ is Riemann integrable on $[a,b]$ and with $P = \{x_0, ... , x_n\}$ a partition of $[a,b]$ and $\forall i\in [1,n-1] : t_i\in [x_i, x_{i+1}]$ then the Riemann sum :
$$S = \sum_{i=0}^{n-1} f(t_i)(x_{i+1}-x_i)$$
Converges for $n\to \infty$ and :
$$\lim_{n\to\infty} S = \int_a^b f(x)dx$$
Now for your example, let's suppose $f$ and $g$ are integrable on $[a,b]$, $S_f$ and $S_g$ are their respective Riemann sum, $c_1,c_2\in\mathbb{R}$, $P_n$ a partition of $[a,b]$ with $(t_i)_n$ as defined above.
If we want to prove that $c_1f+c_2g $ is integrable we need to prove its Riemann sum $S$ converges :
$$S = \sum_{i=0}^{n-1}(c_1f(t_i)+c_2g(t_i))(x_{i+1}-x_i)$$
What allows us to conclude is that we can simply re-write $S$ :
$$S = \sum_{i=0}^{n-1} c_1f(t_i)(x_{i+1}-x_i) + \sum_{i=0}^{n-1}c_2 g(t_i)(x_{i+1}-x_i)$$
$$S = c_1S_f + c_2S_g$$
But we know that since $f$ and $g$ are integrable on $[a,b]$ then $S_f$ and $S_g$ converges for $n\to\infty$ so we can conclude that $S$ converges too, which means $c_1f+c_2g$ is integrable on $[a.b]$. We can even, as a bonus, verify your second point that we already proved :
$$\int_a^b (c_1f(x)+c_2g(x)) = \lim_{n\to\infty}S=c_1\lim_{n\to\infty}S_f+c_2\lim_{n\to\infty}S_g=c_1\int_a^bf(x)dx + c_2\int_a^bg(x)dx$$
