Everyone who says it's tough now, but it will get better, is absolutely spot on, OP. Your baby is 9 weeks old - and you've had a huge shock, both emotionally (the end of your relationship with his father) but also physically (giving birth to your son). You need time to recover, to rest, to deal with the huge swarm of hormones and emotion that you'll still be going through, and to re-adjust to life as a single parent as opposed to being part of a nuclear family with your baby's father. You will recover. It will and does get better. You need to look after yourself as well as your baby - and in many ways, you need to find a way of cutting all of your emotions towards your ex loose. Work out how to co-parent with him by all means... but this is a "man" who, if I've understood correctly, cheated on you with an employee at his place of work whilst you were pregnant and trying to build a future for your baby with him, didn't have the courage to tell you until months later, and has stood by and watched as his family have gone NC with you. Presumably, when he takes your baby to his parents, his girlfriend will be there occasionally - and if my own experience is anything to go by, it'll be his mother caring for your son, not him. Or perhaps even his girlfriend.
When my son was a toddler, his father took him off out for the day, then telephoned me to say that his girlfriend had given birth to their baby that morning, and he was taking our son to meet his new sister. I was frantic as (a) I didn't know if he was intending on bringing our son home again and (b) my family had just been destroyed in one of the cruellest ways imaginable. I found out later that the only reason he telephoned me was because his mother forced him to - she and his father had known about the woman he'd been cheating on me with, for 11 months... they'd looked me in the eye and spoken about what a wonderful man their son was, we'd eaten dinner together as an extended family, my children had spent time with them as doting grandparents. I barely have anything to do with them now. Any of them. My ex pretends our daughter (who was 10 at the time) doesn't exist and undoubtedly contributed to her MH problems, he sees our son for less than 18 hours a month and, then, he's more interested in interacting with the two children he and the girlfriend (now his wife and, actually, a lovely woman whom I get along with as she, too, is being treated in a similar way by him/his parents) have together, rather than ours, and for the sake of my own sanity, I have nothing to do with him anymore. Which hurt more than I can adequately explain as we'd been together since we were 14, having known one another since the age of 11. But... I got through it, and came out the other side a thousand times stronger. Being a single parent is tough, don't get me wrong, but I'd rather that than have my son learn from his father that betraying a partner is the right way to behave, or my daughter believe that it's acceptable for a man to ever treat her the same way as my ex treated me.
You will get through this OP. You're still in shock, still grieving the relationship which you thought you had... except that relationship didn't exist as far as your ex was/is concerned. How could it? He was dating another woman!
If I were you, I'd hunker down with my 9 week old and wait for the hormones to dissipate a little before making any rash decisions about your son having your ex's surname (other posters are right: it will and does make travelling a nervewracking experience if you have a different name to your child, and it might make your son feel "different", or as though he doesn't belong to you, if you have different last names to one another. My children are double-barrelled, and I've never pushed the issue - but both have dropped their father's name, independently, as they've grown older), or even the amount of access which you've given him. A 9 week old baby shouldn't be away from their mum for long periods of time, in my opinion, unless it's because that mother is a sole earner and the only one who is paying for the baby's food/clothing/nappies. Does your ex contribute financially at all, because babies are expensive little creatures!
And if I were your ex's "new" girlfriend, I'd be taking into consideration that if he can happily cheat on/leave his pregnant girlfriend once, he can - and probably will - do so again. Cliches ("once a cheat, always a cheat") exist for a reason.