Best Solo Travel Locations?

$1500 can get you a week pretty much anywhere in the US, if you're willing to do a budget hotel or air bnb. You could also do a Canadian city, or many places in Mexico or Central America. For New Years, places that might be fun (and warm) and cost less than $500 in airfare - New Orleans, Austin, Miami and the Everglades/Keys, the Yucatan peninsula, Bogota, and Panama City (Panama). The remaining $1000+ will go farther outside the US  than in it, but would be doable within the US if you pick a home base in a city that doesn't require renting a car the entire time.

Hello! I would recommend Tamarindo a nice beach town that's not expensive but lively. It's got nice weather with pacific beaches. Puerto Viejo de Talamanca is another good beach town full of young people with a caribbean flavor. I think it's less expensive than Tamarindo. You can stay in a hostel at either and make new friends. Eat local foods which will cost you less. 

Panama City (Panama) where I am from can be fun too. We use the dollar but things cost less. Most young people speak english. The city is very cosmopolitan and you can head to pacific or atlantic beaches anytime. Check out hostels or roommate situations on Airbnb. Again eating local food will save you money and add to your experience. Boquete is a town that you can reach by bus from the city. It's a colorful area full of nature and if you like to hike the perfect location for it. Again look for hostels and eat local cuisine. It has a large expat community.

In the USA, I recommend Rhode Island. I use to go there every summer when I was a teen 'cause my sister's husband's family is originally from there. It's a small state with a lot to offer. Couchsurfing or a hostel a good option. You can have really good seafood here. Check out Newport while there. It's a ritzy area but interesting to walk around and see the architecture. Also if it fits your budget take the ferry to Block Island. I loved going there and walking all over the island :)

Baltimore is another favorite spot for me. I recommend the aquarium and walking about the inner harbor. Look at hostels and couchsurfing. There are good affordable places to eat too. 

Spirit airline is cheap but don't expect anything fancy. Just remember is a way of transport. https://www.spirit.com/  Enjoy your adventure!

$1,500 for how long? Including transport?

You can do something like buy an Amtrak pass. Are you going to use CouchSurfing or other hospitality program. That will save you a bunch of money.

When you are going might make a big difference? For instance, camping saves a bunch of money but you might not want to do that during winter.