I love questions like this. I did a post on team bikes after thinking about this when I joined a new team and they offered bike discounts:
http://sprinterdellacasa.blogspot.co...team-bike.html
For budget keep in mind pedals, shoes, helmet, some clothing, and fit items (bar/stem/saddle). If you don't have a smartphone you probably want a basic computer to track distance etc. Cadence is actually sort of significant at the beginning.
With that in mind remember that the bike has to fit. If it doesn't fit it won't do you much good. This includes bar height/drop, which is critical. You don't want to be too upright in a race, it's probably one of the biggest performance limiters you have control over (genetics you can't control, and fitness at a given moment you can't control either). The Synapse has a higher bar height for a given size. This may be a fit limiter.
The Synapse has longer chainstays. This makes the bike a bit more stable, a little less nimble. This is a minor point, shouldn't be a big thing, but a shorter chainstay bike will feel more responsive when out of the saddle. In the saddle, not really. You can change a lot of things on a bike but you can't change the frame geometry so make sure it's going to be okay for you.
Given the choice between a similarly priced/set-up, say, CAAD8-10, I'd choose the CAAD8, mainly for fit and (minor) handling improvements. Not a deal breaker but for long term, if the fit works, I'd consider a CAAD over a Synapse.
Please budget for fit items. These include saddle, bars, and stem. Again, fit is absolutely critical. If a normal frame works for you then you can transfer the bar/stem from one bike to another. Saddles usually transfer as well.
Having said that remember that your fit will evolve as you become more race fit. It's not like you'll be able to do yoga like stuff on the bike suddenly, it's just that you'll feel more comfortable leaned over a bit more, recruiting more cycling specific muscles currently not developed, etc.
Some of my thoughts on fit (this relates to a teammate I fit who was obviously a bit too upright in his position and cramped fore-aft, even though he was "professionally fit"):
http://sprinterdellacasa.blogspot.co...ive-thing.html
There is a link at the beginning of that post going to something a bit more in detail about fit changes in a new racer/cyclist.
You'll want clipless pedals. The Synapse has some but not ones that you'll keep if you keep racing. A Look type system (Keo, similar) would work well for a long time. You can get them cheap. If you are limited in budget and you get a decent bike that stretches your budget I'll send you a pair of (used) Keo pedals free.
Gearing - really you'll need just a close-ish ratio cassette, maybe an 11-25. That will set you up for most events. The derailleurs should be fine, just adjust the rear to follow the cog profile closer (b-screw or similar adjustment).
Stay with Shimano or SRAM. I say this as a Campy user. Campy replacement cassettes are harder to find and typically more expensive. Historically this has always been the case. With 105 you simply can't go wrong, it's perfectly okay for a Cat 3 lifer. A friend/teammate worked through the ranks from Cat 5->3 on an aluminum 105 bike (Felt). He rewarded himself with a carbon Ultegra bike (Trek). He said it was the stupidest thing he did. He got carbon tubular race wheels ($750 with tires) and he said that was the best thing. He sold his carbon/ultegra bike because he didn't see a true performance value in the rig, and he could sell it for more money. With 105 you'll replace wear items as necessary, then you might go Ultegra (for cassettes, chains). For derailleurs not necessary.
Wheels - yes, they help, but get fit first. Then worry about the other stuff. Decent starter wheels will serve as great training wheels later.
I don't know if this bike was stolen but this would fit pretty much everything except no pedals. No extra money for wheels but those wheels are fine - in fact I have basically the same wheels for me. Tires are great for both training and racing. It's a bit high up front so if that fit works for you then great. Chainstays are short (note small clearance between rear tire and seat tube). Should be pretty stiff/beefy for racing. Carbon fork so front end will be as comfortable as anything out there.
http://baltimore.craigslist.org/bik/5888687549.html
Hope this helps.