How do I find a regular expression for this language, and how do I define a finite state machine that recognizes words in the language (input alphabet, states, start state, state transition table, and accept states). Including a state digraph for the FSM for this language,
L: For alphabet $\{0,1\}$, all strings of length 3 or less that have more $0$'s than $1$'s
You state machine will need to count. In general, state machines cannot count arbitrarily far, since they only have finitely many states, but they can count up to some maximum. In your case, you'll want to keep track of the number of $0$s and $1s$ until the sum of both counts exceeds three. Once it does that, you switch to an "overflowed" state, from which you never switch again, and which isn't an acceptance state.
If $(a,b)$ says you've so far seen $0$ $a$-times and $1$ $b$-times, and $\infty$ is the overflow state, you'll need the states $$ \{(0,0),(0,1),(1,0),(0,2),(1,1),(2,0),(0,3),(1,2),(2,1),(3,0),\infty\} $$ It shoud be easy to describe the transitions between those states - what, for example happens if you're in state $(1,2)$ and the next input is a $1$? It should also be easy to find the acceptance states - just compare the counters for $0$ and $1$ to the condition stated in your question.