A collections of thoughts about a few of the questions you've raised. The short answer to the title question is 'yes'. Read further for some specifics.
If you have the budget for multiple new bikes, that's great. However, based on your apparent wants, I'd suggest the 'more serious rode bike' be a cyclocross bike. You can put slicks on a 'cross bike and it is effectively a road bike. You simply put on different tires for when you want to race/train in the mud.
Winter and Spring Training
If it's in your budget, definitely get a trainer. They are far from exciting, but regular sessions on the trainer through the winter will do wonders for your fitness coming into the spring. For the most part you will want to be riding so that you're sort of breathing deeply, but could still carry on a conversation.
There really isn't a whole lot of point to doing anything really painful during this period, other than combat boredom of long winters. If you're into snow sports, cross country skiing is a great way to get/stay fit over the winter that doesn't involve being inside all the time.
Spring and Summer
Ride lots. Have fun. Listen to your body and get professionally fitted on your new 'cross or road bike, it is surprisingly easy to get injured early in a new sport. You will want to start riding faster every now and then. Find a couple of local routes or hills that you occasionally try to set personal records on.
You are new at the sport, so in most places your races will be < 40 minutes. This means you don't need to go out and hammer for 4 hours. (And shouldn't unless you really enjoy this)
The only real specific training I'd suggest at this point in time (later summer) is occasionally do things like 20 seconds sprinting, 20 seconds soft pedalling, repeat for several minutes at a time. Take a break for a few minutes then do it again a couple of times. Do this once or at most twice in a week.
The major reason I suggest this is because 'cross is extremely stochastic. Unlike road racing (running or cycling), triathlons or most other reasonably common endurance sports, a 'cross race is often 10 seconds flat out, 5 seconds braking and coasting around a hairpin, 30 seconds flat out, get off the bike, run, pedal carefully through an off camber corner, flat out for 15 seconds... You get the idea, but your body needs to experience this a few times before you do it in a race.
While I don't think you need to spend huge amounts of time on this early in your career, you also don't want to try this stuff out for the first time in a race.
Find a nice park nearby. Find some trees that are close together and try riding in an '8' pattern around them as quickly as you can. Do this in dry and wet conditions. Your bike will slip out from under you eventually. Learn what it feels like just before the bike washes out, this is important.
Every race will have a part where you need dismount. Ride for 30 seconds hardish, dismount, run with the bike for 30 seconds, then hop back on and ride again. (Youtube for techniques, I'd suggest not trying the flying remount, it's generally not worth it...)
You've practiced, you've been riding for a year, great! Get out there and race. Do at least a lap or two of the course before the race. You want to be pretty warmed up before you start and then just cut loose.
A couple of other random points. Due to the hard/coast/hard/coast nature of a cyclocross race, you will almost certainly blow up. This can be a new and exciting experience so you should be aware of it. You will want to get off your bike Right Now and maybe lay down for a while. This is normal. Cyclocross is an endurance event, so you will need to occasionally lower your pace and recover.