Is there a general procedure for inducing a formula / recognizing a pattern? For some problems, it's pretty simple to deduce what a formula should look like after you enumerate a few examples, but for some problems, it's not so clear. For these less clear examples, is there some procedure I can follow to find the formula?
For example, just now, I was working on the problem:

Find the expected number of tosses it takes to get $k$ 6s in a row.

The recursive formula is
$$
E[N_k ] = 6(E[N_{k - 1}] + 1)
$$
where $N_k$ is the number of tosses it takes to get $k$ 6s in a row. The initial condition is $E[N_1] = 6$ because $N_1$ is a geometric random variable with $1/6$ probability of success.
Then I enumerated a few cases:
$$
E[N_1] = 6 \\
E[N_2] = 42 \\
E[N_3] = 258 \\
E[N_4] = 1554
$$
It's not obvious to me what the formula should be. But I think it should look something like
$$
E[N_k] = 6^k + (k - 1) * \text{something} + \text{maybe other terms}
$$
But I don't know what this "something" and "maybe other terms" should be. If I sit here long enough I could probably figure it out, but is there some kind of general procedure that I can apply to a wide array of problems?

You asked

is there some kind of general procedure that I can apply to a wide array of problems?

Around 1960 there was some work on a General Problem Solver which had very limited capabilities but much progress has been made since 1960.
Still, there is no way to solve all problems and probably never
will be.
However, the solution of linear recurrences is known for a long time. For example,
$$a_n=6(a_{n-1}+1)=6a_{n-1}+6$$ is easily solved using the theory.
For this particular difference equation with initial value $\,a_1\!=\!6\,$
a lookup of $\,6,42,258,1554\,$ in OEIS returns
OEIS sequence A105281 "a(0)=0; a(n)=6*a(n-1)+6."
This entry has a program

(PARI) a(n)=if(n<0, 0, (6^n-1)*6/5)

which gives a formula for the solution.
The Mathematica program has a function RSolve
which solves recurrences and also has limited capabilities
You can use the Mathematica (or WolframAlpha) command
RSolve[a[n] == 6 a[n - 1] + 6 && a[1] == 6, a[n], n]

to get the particular solution $\,a_n = 6 (6^n-1)/5.\,$
