If you want to outrun a moose, you’re going to need some powerful stimulants. Meth is far and away the best (for emergencies only), but Adderall is also a powerful stimulant, and in a pinch even a couple of caffeine pills might be able to do it especially if you already have Quick and Fleet-Footed.
You can also fight moose if you have the skills or tools–early on this is easiest to do with a strong ranged weapon (a moose encounter in the early game warrants shotgun use if you have one), but once your skills are developed, it’s entirely possible to fight a moose in melee. With developed characters I’ve just shrugged my shoulders and punched them to death before, as well as their zombie variant, the antlered horror.
Some recommendations for actually standing and fighting a moose, that apply to fighting in general, especially in the early game: sewing kits are your friend and reinforcing all your gear will dramatically improve your survivability; Parry is an excellent melee weapon property that can be found on things like baseball bats or even the humble pipe, and it too will protect you from quite a bit of damage; torso encumbrance is very bad mojo for melee as it will significantly increase the time it takes to make a melee attack and also drastically penalize your Melee and Dodge skills, and if they’re penalized below 0 you’ll be a sitting duck (a note: the basic Survivor’s starting winter coat is a lot of torso encumbrance!); leg encumbrance isn’t quite as bad as torso encumbrance but will still penalize your Dodge skill to a lesser degree than torso encumbrance.
Sadly, you can’t customize your Safe Mode warnings. If you know you’re in the presence of dangerous enemies (or just dogs) already and don’t want the game pausing when something new steps in, you can push shift+1 to temporarily turn Safe Mode off–if you have auto-Safe Mode on, it’ll re-enable itself after several turns with no visible enemies. Pay attention to the sidebar when you have Safe Mode off, because you don’t want to fail to notice a hulk or brute or shocker or moose or such showing up.