Yes you can have both an SSD and a HDD on the same PC. They would be on separate SATA cables and ports. And yes you can use an SSD for the OS and applications, and a HDD for storage (very common).
The price of SSD's have dropped a lot recently so you can get one of less than $1 per GB. My recommendation would be to purchase a 256GB SSD (about $200, some less). That way Win-7 or Win-8 plus all your application programs will easily fit on the SSD, and most data files, with plenty of space remaining for expansion. Photos or Music can be stored on a storage HDD as you were thinking.
A 128GB SSD might work, but over time you will fill it up, and will be back in the market for a larger SSD.
And yes they are sooo much faster.