Understand the given functions.

This means $f(g(1))$.
First, evaluate $g(1)$:

$$g(1) = 1 + 1 = 2$$

Next, evaluate $f(2)$:

$$f(2) = (2)^2 + 1 = 4 + 1 = 5$$

So, $(f \circ g)(1) = \boxed{5}$.

This means $g(f(1))$.
First, evaluate $f(1)$:

$$f(1) = (1)^2 + 1 = 1 + 1 = 2$$

Next, evaluate $g(2)$:

$$g(2) = 2 + 1 = 3$$

So, $(g \circ f)(1) = \boxed{3}$.

This means $f(g(x))$.
Substitute $g(x)$ into $f(x)$:

$$f(g(x)) = f(x+1)$$

Now, substitute $(x+1)$ into the expression for $f(x)$:

$$f(x+1) = (x+1)^2 + 1$$

Expand the expression:

$$f(x+1) = (x^2 + 2x + 1) + 1$$ $$f(x+1) = x^2 + 2x + 2$$

So, $(f \circ g)(x) = \boxed{x^2 + 2x + 2}$.

This means $g(f(x))$.
Substitute $f(x)$ into $g(x)$:

$$g(f(x)) = g(x^2+1)$$

Now, substitute $(x^2+1)$ into the expression for $g(x)$:

$$g(x^2+1) = (x^2+1) + 1$$ $$g(x^2+1) = x^2 + 2$$

So, $(g \circ f)(x) = \boxed{x^2 + 2}$.

This means $f(g(a^2))$.
First, evaluate $g(a^2)$:

$$g(a^2) = a^2 + 1$$

Next, evaluate $f(a^2+1)$:

$$f(a^2+1) = (a^2+1)^2 + 1$$

Expand the expression:

$$f(a^2+1) = (a^4 + 2a^2 + 1) + 1$$ $$f(a^2+1) = a^4 + 2a^2 + 2$$

So, $(f \circ g)(a^2) = \boxed{a^4 + 2a^2 + 2}$.

This means $g(f(\sqrt{a}))$.
First, evaluate $f(\sqrt{a})$:

$$f(\sqrt{a}) = (\sqrt{a})^2 + 1 = a + 1$$

Next, evaluate $g(a+1)$:

$$g(a+1) = (a+1) + 1 = a + 2$$

So, $(g \circ f)(\sqrt{a}) = \boxed{a + 2}$.