Do we need to move it into a bigger pot and what kind of fertilizer should we use?

Brown tips occur for different reasons including dry air, fresh tap water or the soil is kept too dry.
Since your PL is in/near the kitchen, cooking stoves/ovens dry air.

Can you move your PL in a different location, away from dry heat?
Another option is by allowing tap water to sit overnight before giving your plant a drink. Sometimes chlorine causes brown leaf tips.
Although I let soil dry a wee-bit between waterings, when you water, saturate the root-ball entirely, until water seeps out of drainage holes.

For the time being, if any of my suggestions apply to your PL's brown tips, please correct them.
Cut brown tips off but leaves about 1/8" of brown on leaves. If leaf is cut into green, browning can spread.

It's a good idea to either mist or shower foliage regularly. Keeps leaves free of dust, helps prevent insects, and increases humidity. Good luck, Toni

My peace lily leaves are just starting to brown, I think it may be the tap water.

It's true misting up's humidity for short periods, but if plants are grouped together, a non-generic sprayer is used, 'I prefer a mister w/hose,' weekly showers, and of course a humidifer, spraying makes a huge difference.

When I was ill, 'more than once,' I neglected spraying. Man oh man, what a difference! Even if spraying is short-lived. I try spraying 1-3 times per day. Excluding succulents/cacti.

Is the brown on your PL crispy or soft?

It could be the water, which is one reason I keep old milk containers filled w/water, ready to use,' or now that winter is approaching air is dryer.
Humidity of late has been low; at least in IL. Toni

My grandmother doesn't think the water is part of the problem, since we live on a well, but it's entirely possible our water is too mineral rich for the plant.

The brown on the PL are crispy, but a few new leaves are popping up so I don't think it's close to giving up yet.

If you want to get a junker car to run, do you first replace the missing motor or fix the transmission that won't shift into high gear? You should do both, of course, but replacing the missing motor is going to move you closer to your goal than fixing the transmission and is the logical focus. It's the same with your plant - make sure your roots are healthy and have the oxygen they need to function. Saturated soils, ie soils that support a lot of saturation, reduce the oxygen in the soil that roots need to function. This creates lots of problems, one of them being spoiled foliage.

You might benefit from reading the link I left below. It gives an overview of some of the considerations it's prudent to take into account.

Take care.

Al

I must confess, there were times my Spider Plants weren't fertilized or were fed with Fish Emulsion. 'Organic Plant fertilizer,' yet tips turned brown. I'm talking about times I watered directly from the faucet, instead of using water that sat out.

Even if Chlorine doesn't make a difference, 'although there are numerous plant book authors who vow chlorine will brown tips, especially amongst Chlorophytums,' IMO, room temp water is best. No shock from too cold or hot water.

So, whether it makes a difference or not, I feel comfortable using room temp water that's been sitting 24-hours or more.

Watering with room temp water is not written in stone, however it is written in plant books. And by plants' appearance.

Just my opinion, for what it's worth. Toni