Hi Andrea! I am from The Netherlands. I have done several bike trips in the US, including the TransAm. From your question I guess that you are new to biking in the US.
Here is some advice.
1. In any case, take an ACA mapped route. This will save you a lot of time in navigation and keep you away from awful, busy roads, e.g. Interstates. The maps also provide information on services ahead, which are vital in sparsely populated areas. You will also meet other bikers going the same way.
2. If you want to start in San Francisco, the only ACA route to the east is the Western Express. It connects to the TransAm at Pueblo-CO. As others have commented, it is exceedingly hot in summer. Compared to the full TransAm, it saves about 500 miles, but your riding hours will be less too because of the afternoon heat. Buy the ACA maps asap to make up your mind. They give you an idea about the distances you should cover each day to get to required services. Maps also have information on Climate, e.g. temperature statistics.
3. The full TransAm took me 60 days, including rest days, in May/July. This amounts to 80 miles/day. With your time plan, you need to cover 80-90 miles/day. To get a more relaxed scheme, you may consider to start your bike trip in Denver-CO.
4. Think well how you want to travel with your bike to New York. When you follow the TransAm to the east, you get to Richmond-VA from where you can take a train to Washington and New York. However, Amtrak requires to box the bike. A boxed bike is nice for air travel, where ground personnel does the handling, but it is a millstone around the neck for rail travel, as you have several other bags to care for. It is easier to send the bike home by mail, but this is a lot more expensive than the charges of airlines for bike transport (I got quotes of about $600).