• Your employer should give you a rundown of benefits during new employee orientation, etc.
• One of the benefits that many employers offer is pre-tax transit account.
• When you get paid, you get taxed on the amount you get paid every paycheck. Example: Get paid $1,000, tax withholding is a total of 25%, you get $250 withheld from your paycheck (note: you may get some of this back when you file taxes and get a refund, but for budgeting purposes, you're only taking home net $750 every pay period despite gross $1,000 pay).
• Things like insurance, transit accounts, etc. are pre-tax, meaning they are deducted from your taxable income. Example: If paid $1,000 and your pre-tax benefits cost $150, then you're taxed 25% only on $850 instead of $1,000. Basically, the money you set aside each paycheck for whatever benefit is 'diverted' before taxes apply. So no taxes apply to the $150 and it goes straight to whatever the benefit is (health savings account, transit account, etc.) You essentially take home an extra $37.50 per paycheck to 'spend' on transit, healthcare, etc. instead of the government withholding that (until you get a refund).
• The upshot is that you now have money to spend on transit fares and you didn't have to pay taxes on the money you earned. It's 'taken out' of your paycheck and put into a separate account that you have access to and can only use for transit expenses, but you were going to spend the same amount of money on transit anyway and now you saved some money by paying less in taxes overall.
• I have wageworks for transit benefits. Whereas I have to choose my health insurance, retirement when I first start my job, I can opt into wageworkers anytime. Once I opted in, wageworks sent me a debit card that I can use for transit purchases. I chose to divert $X every paycheck to my wageworks transit account, and that debit card lets me spend that money I accumulate in the transit account. It's all money that I earned but on which I avoided taxes.
• You can use the debit card at the PATH or MTA machine, and/or add the debit card to your OMNY account and connect the OMNY account/card to your contactless payment (Apple Pay, Google Pay, whatever).

For PATH trips, I recommend that you get a SmartLink card and buy bulk trips (40 trips) with auto refill. You get a modest discount when you buy bulk. The monthly unlimited 'pass' (which is loaded onto your SmartLink card) is only worth it if you're taking PATH everyday and taking weekend trips into the City, otherwise don't bother. For MTA, like others have said, if connect to OMNY and your phone, your payments per week are capped.