The hardest thing in my opinion is dealing with unreasonable customers, who can take many forms.
Biggest tip I can give you is to ask questions both during and after being trained. If you don't know something with 100% certainty, ask. If you're wrong, we'll have to set you straight later anyways, and possibly while fixing something complicated with an irate customer.
Wic? Ask. Special sales? Ask. Tax exempt, returned checks, coupon error messages? Ask.
And don't ask the customer questions about procedure. Best case, they don't know. Worst case, they could lie to gain an advantage.
Soul crushing boredom is a big problem. Another problem is having to deal with idiots and jerks all day long without being able to jump over the cash register and slap some sense or manners into them.
As a new cashier, you will specifically be targeted by extreme couponers hoping to take advantage of you. If you ever feel like you are in over your head, call for a manager.
The customer is almost always wrong about absolutely everything. The real question is, when should you just go along with it?
The cash register is fairly straightforward to operate once you figure out the basic syntax. Until you tender out the order, pretty much everything can be fixed by you without calling for a manager. Except for voiding produce. You will have to call the manager half the time for that.
the most important thing is keep track of your money.
always count what the customer hands you, don't take their word for it. tell them what they gave you like if the total is $11.21 and they give you $20.25 tell them it's 11.21 out of 20.25. Some will say "i have 21" and give you a quarter or whatever so don't cash out until you have their money.
count back their change and give them exactly what the register tells you.
do not break bills or make change. they will scam you in a heartbeat.
make sure you are doing id correctly as the training class tells you. usually there's some restrictions like vertical id has to be put in by a store manager. don't put random dates in. we've lost cashiers that way and it could have legal consequences for you and your store if you sell to underage.
if you're not sure you can accept an id because it's out of state or foreign or whatever, ask your floor supervisor or manager.
on tobacco, if your store doesn't have a specific register for it don't hand it to the customer until they pay. same with gift cards. hold those until they've paid.
don't clear a BOB (bottom of bascart) warning without knowing the item or items were scanned.
It's not hard to keep track of money, but a lot of employees do struggle with it. Always count the customer's money, and tell them how much they gave you. Usually, you can do this when you tender out the order or if they are having trouble counting it out.
Be careful when you count out change. I like to count it twice because bills often stick together or behave unexpectedly. Customers sometimes fold their bills in obnoxious ways and often can't understand wallets, so it happens more than you might think. If it's a large amount, you might want to count it out loud for them. If it's a small amount, I like to fan it out so the customer can hopefully count it at a glance.
Change scammers got me once or twice. One of the times, it was when I was cashing a CoinStar, so it can be difficult to avoid. When bills start going back and forth, it's easy to get confused. Insist on doing things one step at a time, and you might be ok.
Checking ID's is Serious Business. Cashiering at Kroger is bad enough. You don't need jail and fines. If they look younger than such and such age and don't have ID, too damn bad. In my state, you pretty much need to have ID anyways, so not having one is suspicious. When you decline the sale, ask them if they'd like to speak to your manager. The manager will have your back because it is so serious.
Learn your produce. Take a walk through produce when out of uniform. You can't look it up if you don't know what it is. Customers often don't know what they are buying. Either it's an expensive tropical thing that caught their eye, or they are lying to your face since you might give it to them cheap.