My son has a viral infection, could I get sick?

So, I took my son to the dr yesterday because he kept gagging on mucus when when he would cough and his nose is running clear. He is 17 months old and has never been sick so I kinda freaked and took him in. But she said it was a viral infection (he started with the symptoms on Christmas morning, which was a congested coughing) and she gave him some antibiotics but told me not to get them filled until 10 days p and he is worse. But she said she didn't want him to get started on them until he was alot worse than what he is now and I would to because I really don't care for pumping him full of drugs he really don't need. Anyways, my question is... can I get what he's got? I let him get a drink of my drink and then I got a drink right after without thinking I could get sick too. But is he is contagious??