See [here](http://stats.stackexchange.com/a/77407/36229): in general, the error process only needs to be _homoskedastic_ over time.

If the model was fitted with a Gaussian likelihood, you do need to assume normal residuals to conduct parametric inference, as mentioned [here](http://stats.stackexchange.com/a/79793/36229).

It's been a while since I've studied ARIMA in detail, but I suspect that heavy-tailed residuals will cause the same kinds of inference problems for ARIMA that they cause for other regression models that assume normality.

Since you don't say anything about heteroskedasticity, I'm assuming you don't have any. In that case, there's not much GARCH will do for you. GARCH models the change in variance over time, but there's no change to model.

As for modeling in R, "analysis" is a broad word. If you mean that you want to generate predictions, note that adding a GARCH model on top of your ARIMA model (i.e. two-stage estimation) _won't change your forecasts_; but if you want to, say, generate confidence bands around your prediction, you can use the GARCH predictions as the standard errors.

As pointed out in the comments, fitting both models simultaneously is better than fitting one and then the other. I haven't done this personaly with R, but there is a package called `rugarch` might have this functionality. There's a writeup at [the author's blog](http://unstarched.net/r-examples/rugarch/a-short-introduction-to-the-rugarch-package/).