Most babies that age wake multiple times a night and napping is often infrequent or inconsistent at that age because babies don't really start to organize their sleep until around 4 months.
I had a very strong (though not necessarily "precocious") baby. She was holding her head up and moving it independently at birth. But she would not tolerate being put down, whether in a swing or bouncer. We got very good at babywearing. She napped in the sling or the moby on long walks (which were also great for stimulation). But generally, a baby that young doesn't really need a lot of stimulation. Sing to him, talk to him while you do your own work, and spend time outdoors (gardening, walking the dogs, whatever) and he'll get all the stimulation he needs.
The first three months are REALLY intense. I was at my wits end too many days just trying to deal with the huge change that had happened in my life and come to terms with the fact that my body and my time were no longer my own. It gets a lot easier. For us, night sleep got much better at 6 months and day time sleep started getting better around then too (though it was hit or miss until 9 months).
And it helped me to think of those first three or four months as the fourth trimester. Babies are not and cannot be independent at that age. They are 100% dependent on you for everything. It's not their fault, they're not trying to be difficult, and you can't spoil them. So responding to my daughter's needs became a loving act setting us up for an independent future, rather than an exhausting one. It's not easy though. Try to get help if you can from your partner (if you have one) or from family, friends, or other caregivers. Moms need support and help to parent. Motherhood has traditionally not been a solo occupation. You say you don't have generational support, so you might have to go out and create your own. But being around other mothers with babies that age can give you a lot of perspective. And it really does get better! By 6 months when you go back to school, chances are good things will be much different. My daughter was a totally different baby at 3 months than at 6.
Good luck! And let us know how things go.