Welcome to cruise-chat.
If you look here
you find info on the "Royal Tots and Babies" and "Adventure Ocean" programs. The Royal Babies program is a new feature and 6 month olds can attend. Bear in mind you or another adult in your group must stay with the baby. Your older child can be enrolled in the Adventure Ocean program for his age group.
I'd be lying if I told you cruising with a baby is all sun and fun. You'll need at least an umbrella stroller, as well as all the other baby stuff. However, you can now order baby food, diapers, and wipes from Royal Caribbean. These will be in your cabin when you board. Read more about "Babies 2 Go" as well as the Royal Babies and other youth programs in this press release
Nobody can predict the weather in October, or if the seas will be rough. Too many variables.
This is your first cruise so I strongly urge you to not book it on your own. Find a travel agent who specializes in cruises. You'll be glad you did. If you go to the CLIA Website
and enter your zip code on the right hand side, you can find a list of qualified agents in your area.