To solve the expression \( S = 1 + 5 + 5^2 + 5^3 + \ldots + 5^{1501} + 5 + 5^2 + 5^3 + \ldots + 5^{150} \), we can first break it down into two separate series.

1. The first series is:
\[
1 + 5 + 5^2 + 5^3 + \ldots + 5^{1501}
\]
This is a geometric series where the first term \( a = 1 \) and the common ratio \( r = 5 \). The number of terms \( n \) is \( 1502 \) (from \( 0 \) to \( 1501 \)).

The sum of a geometric series can be computed using the formula:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
Plugging in the values:
\[
S_1 = 1 \cdot \frac{5^{1502} - 1}{5 - 1} = \frac{5^{1502} - 1}{4}
\]

2. The second series is:
\[
5 + 5^2 + 5^3 + \ldots + 5^{150}
\]
This is also a geometric series where the first term \( a = 5 \), the common ratio \( r = 5 \), and the number of terms is \( 150 \) (from \( 1 \) to \( 150 \)).

Using the geometric series formula:
\[
S_2 = 5 \cdot \frac{5^{150} - 1}{5 - 1} = \frac{5(5^{150} - 1)}{4} = \frac{5^{151} - 5}{4}
\]

Now, combining \( S_1 \) and \( S_2 \):
\[
S = S_1 + S_2 = \frac{5^{1502} - 1}{4} + \frac{5^{151} - 5}{4}
\]
This can be written as:
\[
S = \frac{(5^{1502} - 1) + (5^{151} - 5)}{4}
\]
Simplifying the numerator:
\[
S = \frac{5^{1502} + 5^{151} - 6}{4}
\]

Thus, the final answer is:
\[
\boxed{\frac{5^{1502} + 5^{151} - 6}{4}}
\]