How to determine whether a function is concave, convex, quasi-concave and quasi-convex

The second is neither convex nor concave - that's easy to determine simply by looking at it. Along the line $y=x$, it is convex as a 1D function; along the line $y=-x$ it is concave. It is neither quasi-convex nor quasi-concave: to show not quasi-concave, consider the points $x = (0, 1)$, $y = (-1, 0)$, so $f(x) = f(y) = 0$. Parametrise the function along that line segment by $\lambda$; then $f(\lambda) = \lambda (\lambda - 1) < 0 = \min \{ f(x), f(y) \}$. You can rotate to get non-quasi-convexity.

The first is convex but not concave, and it's not quasi-concave. Examine the value of $f$ at the points $x=1/3, x=10, x=1$ to see that it's not quasi-concave. It's convex again by inspection or by showing that its second derivative is strictly positive.

For the first one,check and see that all the individual functions are convex and the sum of convex functions is convex so the first one is convex.

Also for the second one you can check along lines as illustrated