With little ones, a lot of times now they will have two nurses do the shots so that baby only really feels a stick once because they both do a shot at the same time. if they don't, you can always suggest it...its your baby and there are plenty of nurses there and it's not at all unreasonable.
Vaccines are very safe, and so beneficial long run when you remember that you're potentially saving your child from having to go through an often painful sickness. I wouldn't seperate the vaccines because that could throw off his scheduling in the future (boosters and series) and also, just getting everything done at once is sometimes just better for them then putting them through that needle stick again and again. Adverse reactions of course is a little pain/soreness at the injection site which doc will usually suggest some infant tylenol. (they updated the mg per weight for infants so ask your doc for a chart to hang on your fridge for future reference and/or ask pharmacist for correct dosage as well because it won't tell you on the tylenol boxes since they're so young). also fever is very common...but here again the tylenol will work.
I would bring along a bottle (bm/formula) or if you breast feed ask if you can sit in the room for a little while (most places will encourage this) and feed LO for comfort.
They actually already have combination vaccines which save our littles ones from excessive multiple sticks. At his two month he should get his first dose of 5 Dtap (diptheria, tetanus, and pertussis), the first of the Hepatitis B series, Polio, Hib, Rotavirus, and pneumococcal disease. Hepatitis B and Hib can be given together and so can Dtap, polio, and Hep B. Combo vaccines are safe and effective and getting several vaccines at the same time will not harm your baby.
At our 2mo, little guy got three shots, as well as an oral solution which he had no problem taking.
Just be on the look out afterwards for allergic reactions, which are rare. LO may be fussy and irritable more than usual and I do suggest getting some baby tylenol for him to help with his discomfort.
You're his mama and you're doing whats best for him. Watching him get shots is difficult but just be there to comfort him and know that you're just looking out for him and trying to keep him safe.