fire extinguisher selection & sizing for home use
Unless you have a really clear idea what to do with one, leave the fire axes to the folks who do know what they are doing with one.

When in doubt, get out. Stuff can be replaced. Dead people can't. Killing yourself by trying to fight a fire is bad enough - if you manage to kill firefighters who go in looking for you, it's worse. If it's small and you can apply an extinguisher without undue risk to yourself, only then try putting it out. Remember that most people are not "burned to death" in a fire - most are asphyxiated.

I shade a bit larger (5 lb., or 2A, 10BC) on what I think is an OK minimum than @mjohns615, but otherwise agree, pretty much. ABC dry chemical is safe to use on any home fire, so you don't have to figure out which extinguisher to grab when you are probably a bit panicky and time is of the essence. I'd rather have more than I need than less, so long as it's not to the point of absurdity. I think the one in the garage/shop is a 10 lb.

The US A/B/C ratings are for a "size of fire that can be put out" for each of the "types" (A being paper/wood, B being oils/fats/grease, C being electrical) and the B/C rating is nearly always 5x higher than the A rating for dry chemical, since dry combustibles have embers which the other two types of fire don't, so they are prone to relighting themselves.

While terribly old fashioned, if you can make space in your life for it, the bucket of sand can be quite effective, and the fire blanket can also. In the kitchen, a box of baking soda and/or salt can also be useful (as is the lid of the pan for a grease fire, if it's handy and you act quickly.) There are also "class K" extinguishers specifically for kitchen grease fires (what we'd ordinarily consider class B) but that's more of a commercial kitchen application than most home kitchens.

Once you have extinguishers, you need to keep an eye on them - most have a gauge to indicate if there's a leak, and if there is a leak, they need to be serviced (mostly for the initially-more-expensive extinguishers) or replaced (most cheap extinguishers.) Likewise, if the extinguisher is used at all, it needs service or replacement (the valves don't seal once powder has gone through them - also the contents are not all there.) You can do the inpection yourself (most homeowners) or you can hire a company (most businesses) to do annual inspections and servicing as needed. If you have a serviceable extinguisher, have it serviced (when needed) by a specialist company.