Grab one at the airport. Usually the guy selling you the SIM card will also sell you an EasyCard for $100NTD.
Can be used for purchases at all 7-11/convenience stores, MRTs, buses, Youbike stations, even some other random shops.
Getting the AllPass is also recommended if you're staying longer than a few days
AllPass: includes all MRT rides, most bus rides (within Taipei). If you're using public transportation as much as I did then it's definitely worth.
You can get it at any non-airport MRT top-up machine.
Cash and credit cards
You will definitely need cash - I have a debit card with free withdrawals worldwide (Schwab), and any convenience store has an ATM. Works perfectly.
Generally any mall-like store or actual storefront will take foreign credit cards, but most street stalls and many stores will only take cash. Would recommend getting a coin pouch. If you have too many small (<10NTD) coins, you can dump them into your EasyCard at a topup station.
If you ever get the option of paying with USD at a cash register, say no. The exchange rates are always going to be worse than doing it with your bank.
Data
Get an airport SIM - actual unlimited data for up to a month, all at reasonable prices
I was able to tether to my phone most of the time and watch streams/videos/do work, which makes internet access incredibly easy
This is only for tourists, so if you're staying for longer then this won't work for you
The phone number might show up as a 9-digit number on your iPhone, e.g. +886 9xx-xxx-xxx. There's a hidden 0 in the area code, so you'll want to use 09xx-xxx-xxx instead.
Transportation
Local
I generally prefer MRT > biking > bus > taxi
MRT is great, just follow the (English) directions in each station
Transfers are generally painless (except to the brown line), and you don't have to worry about looking at the schedule (there will always be a train in <10 mins, usually <5)
There are luggage lockers in pretty much every station. Typically you can store stuff in them for up to 2-3 days (around $10NTD/hour), but I found them most useful between lodging (like storing stuff after checking out of a hotel and before leaving). Large stations will usually have a camera pointing at them, so it feels quite safe.
First 30 mins are free, $10NTD every 30 mins after, and more expensive if you keep for longer than 4 hours
Note that you can literally just grab one, put it in a rack, then grab it again to reset the free timer
There are lots of dedicated bike paths in the city.
I did not try this option until very late in my trip, which I regret.
Buses
Generally only useful if you're traveling where MRT and walking is inconvenent (e.g. from Taipei 101 to Raohe)
Would NOT recommend, since they're often not on time (traffic) and can also be less frequent than MRT, unless you're traveling across the city (e.g. to Yangmingshan or the Palace museum)
Tap on and tap off with your EasyCard, or you'll get charged the max (if you don't have the AllPass)
No buses after ~12AM
Taxis
Generally around everywhere, decent option if it's late at night and MRT isn't running
If you can't read Chinese, know that if there's a lit up sign on the taxi (空车) then it's empty
If you're not going to a well known place (e.g your hotel), then just tell them the closest MRT station or other landmark. Less miscommunication that way!
Thankfully not too expensive. Allegedly Uber prices are much higher than a local taxi though, so don't use it.
Long distance
Taking the HSR is convenient, and there's a discount for foreigners ONLY if you prebook online (iirc on Klook?)
I found a lady selling $100NTD bento boxes at Chiayi station (next to the tickets). Would recommend trying.
If you want to take the train to Taroko/Alishan you have to book ahead.
Long distance buses will sometimes require you to book ahead/buy a ticket at the station, e.g. traveling to Yilan. Don't get caught empty handed, get there ahead of time.
You can recharge your EasyCard, get cash, get alcohol, and get food (24/7)
If you get heatable food the cashier will offer to heat it for you. Say yes.
Every convenience store is slightly different in what snacks they offer.
If you're staying for a while you can grab a XL jug of water (6L) for $60NTD.
Look at the combos/discounts offered - you can get a sandwich + drink for ~$75NTD, which is a decent meal.
Personal favorites: Try the fruit cups, and inspect their selection of chuhais (alcoholic sodas)
There's a particular ramen brand that has actual beef in a packet (滿漢大餐) - try it out. Use the free hot water and eat it in the store.
Things I haven't tried but heard were good: tea eggs, heatable burgers
Claw Machines
These are everywhere, usually $10NTD per try. According to the internet, this is due to easy maintenence + easy profit + small investment (you can buy a single machine and rent a place to put it)
There are several interesting claw machines out there! I've seen ones that are custom minigames (e.g. roll a couple of dice, drop the ball onto a holder), unusual prizes (random lost packages, miscellaneous snacks, laundry detergent, etc.).
These claw machines are programmed to grab the prize, then toss it to the upper right corner. Only on occasion (this is a % chance set by operator) will the claw actually keep hold of the prize. Don't count on just grabbing it to work.
The correct strategy is to make the claw somehow push the prize into the hole, or create a slide that will let the prize slide into the hole when it gets tossed. Swinging the claw before dropping it can sometimes make it toss the prize in a better direction. Watch the locals do it.
If a claw machine looks like it's too good to be true, then it's bait. Taiwanese operators are extremely skilled at making a prize look easy - the best bait I've seen is a machine that taped down 2 toys on the side of the hole, holding up another prize right on the corner of the hole... which is impossible to push in. Another oddly specific bait I've seen is a coach bag that has 80% of the body over the hole. (The trick is that the bag is heavily weighted by sand inside so you can never knock it in).