I'm new to freelancing and I'm wondering what, if any, would be a polite way to tell a client that I've got clients ahead of them and that I can't start their project till a certain date?
I'm currently booked solid until $Date. I hope that works with your time frame.
The client will hopefully be OK with that.
When you write your contract (you should be using written contracts for ALL your work), you can put a start date (or tentative start date) and estimated completion date (and milestone dates, if applicable) so it's not a surprise for the client.
Consider a save-the-date clause in your contract. From slides 107-109 of Dan Mall's presentation, "Prototyping Your Business":
"I'll be honest: my time is in demand. To reserve a specific timeframe for your project, you can opt into an additional retainer that guarantees that your project is my first priority starting $Date. This retainer is non-refundable."
Then, there's a checkbox:
"I choose to immediately reserve this timeframe for my project and will send $____ along with the signed contract."
This assumes that the client really wants to work with you, and that your services are in demand enough that you need to do this.