Is it typical for breast/chestfeeding to hurt a bit?
Maybe a bit at the start while you and baby get your bearings but ongoing, no, it shouldn't hurt
When feeding for the first time (or when lacking some fundamental basics of feeding) the first few days after baby arrives is often an "adjustment period." Some experience soreness and can be raw or even mildly abraded at the nipple and/or areola.
This happens for a couple of reasons:
You're learning how to feed a baby, often this makes us readjust our latch a lot until it feels just right. This frequent adjustment involves bringing infant on and off the nipple a lot, which can hurt.
Colostrum is harder to remove than milk that has fully "come in" and requires a lot of sucking at the breast/chest because such small amounts are transferred. So there is a lot of friction and pulling at the breast while you simultaneously learn to feed.
The latch might not yet be optimized and baby’s forceful suck may be directed to the wrong area of the nipple or areola. This might be a cue to get a support from an experienced feeding parent in your network, or from a professional.
"Beyond those early days where you and baby are figuring out how each other works and after your engorgement (if you get some) settles down…you shouldn’t be feeling pain."
The first stage of milk production creates colostrum, super sticky, antibody rich, translucent yellow colostrum. Because of the viscosity of this first milk, it can take longer periods of sucking to draw this type of milk out of the breast/chest. All of this friction at the breast on a potentially misdirected latch can leave you feeling tender, especially if you are not quite sure how to help baby latch properly at this point and it isn't feeling right.
When the feeding dyad is first figuring out their "feeding dance," or whatever you want to call it, there can be a lot of coming on and off of the breast, as mentioned. This on and off, and on and off can injure the areola and nipple unless you first break the suction between your chest and baby's mouth. Babies use strong vacuum suction as a means to remove milk. If you pull an infant off of the breast/chest without first breaking that literal vacuum seal your nipple will be pulled by baby and they will likely clamp down harder and apply pressure at the tip of the nipple (very painful).
How to break suction when taking baby off to prevent pain and possible injury from the get-go
Hook one of your (clean) fingers inside the corner of baby's mouth, pad side up
Place your finger between baby's gums and rotate your finger a bit or apply some gentle pressure upwards with your finger to break the suction.
As you break the suction use the arm supporting baby's arm/back/shoulders to move infant away from the breast so that you can take a second to catch your breath and try latching again more comfortably.
Aside from those early days where you and baby are figuring out how each other works and after your engorgement (if you get some) settles down you shouldn’t be feeling pain. The physical sensations from feeding your child should not be causing you distress. If this is the case you deserve some additional support from your community to achieve your feeding goals
If you're in pain without support, you're at risk of stopping breast/chestfeeding, because it just won't feel pleasant. "For every day of [parental] pain during the first 3 weeks of [chest/breastfeeding] there is a 10-26% risk of cessation of chest/breastfeeding" (Schwartz et al, 2002). If your feeding plan included breast/chestfeeding and you're in pain in the early days and can't quite figure it all out---there IS support out there such as public health lactation clinics and tele-support, private consultants that come to you and a great deal of helpful online videos. These resources may take you from dreading feeding to welcoming it, because it doesn't hurt.
Having said that, if you're in pain and you know that in your heart of hearts this whole breast/chestfeeding thing just isn't for you, and it's maybe even affecting your mental health, it is BEYOND OK to step away. You don't need anyone's permission. A happy, fulfilled parent is the most nourishing thing possible for a babe.
Other stuff that might be causing you pain:
baby's oral anatomy is such that feeding may require specific support, e.g. tongue-tie, cleft palate, etc.
baby is hypertonic (the opposite of floppy) which appears as baby's body being rigid with decreased mobility. Infants with hypertonia often arch away from breast/chest and tend to clamp down when anything is put in their mouth. They may have a tense jaw and restricted tongue movements. Hypertonia can be caused from injury at birth and requires assessment from a medical professional immediately.