(1) Just cause you get a standard power grid up doesn't mean you should get rid of your generator. If storms are really giving you fits keep a genny by your response center; this should ensure you can send out repair orders to everybody to work quickly together, although taking one yourself will be fastest.
(2) "Save money for a rainy day." Literally. Part of the challenge of management here is not spending every penny you earn the moment you get it. Don't just rush through the tech tree: take 5-10 minutes to get $750k-$1M and then don't spend yourself under that for the rest of the game. I don't strictly follow this rule myself, but if you're really struggling that is my suggestion.
(3) Listen for clues: You will hear a clue twice as long before the storm comes as when you get the official warning on your HUD. You will hear the rumble of distant thunder, and if you are zoomed in close to a building you will hear the park PA announce a weather report, and if you are driving a vehicle you will hear a ranger call out a storm on the radio. I don't know the exact timing, but it feels like these clues arrive 20-30 seconds before the storm, while the official warning on your HUD pops up 10-15 seconds ahead of time.
(4) As soon as you get the official warning hit pause, drop whatever you're doing, and open all shelters. If your coverage sucks, hitting it sooner cause you listened and heard a clue helps.
(5) Recall ranger vehicles, especially during snowstorms. Cancel ranger patrols, and try to get them out of the paddocks before the storm hits. Doing this as soon as you hear a clue helps give you more time to get this done, but this is lower priority than opening shelters- see (4). Leaving a ranger in there further upsets the dinosaurs and can increase the likelyhood of attempted breakouts, to say nothing of the trashed ranger who can't escape in the snow. This does not apply to the MVU, because it's job is critical and (at least for now, just in case it's a bug) it can still move around in the snow and get out when it's done.
(6) Repair broken fences, attractions, power stations, transformers, amenities, and then anything else in that order of priority. If you can't afford it, refer to (2). This should allow you to get your income rolling back in even if you can't afford to repair everything.