If you want just some basic ideas what you might test.
The simplest test just generates a million random numbers and puts them into say 100 buckets depending on their value. Each bucket should contain about 10,000 random numbers. If not, your random number generator is off. However, just generating 1, 2, 3, 4, 5, ... will pass this test, so it's not a very good test.
Next, generate a million "random" numbers. Count how often the second of two consecutive "random" numbers is larger than the first. This should happen in about 50% of all cases (the very first random number generator that I ever used had a distribution of 40% / 60%).
A slightly more complicated one: If you generate a, b, c there are six possibilities how these numbers could be ordered. Each possibility should happen 1/6th of the time. If you examine Marsaglia's XORshift generator where he programer has to choose some constants for the generator, quite a few constants fail this test.
Each test that you pass gives you a bit more confidence that the random number generator is Ok. Once you run out of ideas for more tests you follow the links in Mars' answer. I wouldn't worry about a set of tests being 10 years old, there's not that much changing. The examples that I gave for example are nowhere near state of the art, but you'd still have to pass them, like you'd have to pass the TestU01 tests.