Chernoff-type inequality for random variable with 3 outcomes
You can get fairly good bounds by considering the random variable $Y_{ij}$ which is 1 if $X_i = j$ and zero otherwise (for $1 \le i \le n$ ranging over trials and $1 \le j \le 3$ ranging over categories). For any fixed $j$ the $Y_{ij}$ are independent and therefore $\sum_i Y_{ij}$ can be analyzed using Chernoff bounds. Then do a union bound over $j$.

If the above isn't enough I suggest that you look at the balls and bins model e.g. in Upfal and Mitzenmacher's textbook. That model is the same as yours except that some of your bins may be more likely than others to have balls land in them, right? There are some more sophisticated techniques involving Poisson approximations in that model that would likely be extendable to your setting with non-uniform bin probabilities.