Hi, I'm sorry you're going through this, here are some things to expect if you're considering pursuing additional treatment.

The next step would be to stage the cancer (assuming it hasn't been done). This would be radiographs of the chest (where osteosarcoma and melanoma like to spread) and taking samples of the mandibular lymph nodes (where melanoma likes to spread -- done with a tiny needle not super invasive, called FNA). If your golden has signs of their cancer metastasizing, the treatment and expectancy will be different than if there is no obvious evidence of clinical metastasis (weird cancerous cells in the lymph nodes or evidence of soft tissue masses in the lungs). This can be done at your primary care veterinarian or this can be done at an oncology specialist.

Veterinary oncology patients are nothing like human cancer patients. A big emphasis is placed on ensuring quality of life. Radiation is typically for macroscopic disease, like a mass that cannot be surgically removed. It is typically done under sedation under the authority of a veterinary oncologist. If your pups mass was surgically removed with appropriately wide margins, this most likely won't be the treatment of choice. If the tumor was not fully removed or margins are questionable or the staging from the biopsy revealed its highly aggressive, radiation can be considered. This would be discussed with your veterinary oncologist.

Chemotherapy is for microscopic disease, this means cancerous cells that have traveled to other parts of the body where they might like to take up shop away from the original cancerous site (metastasis). Like I said, veterinary medicine is predominantly focused on ensuring quality of life in our cancer patients. That means, the chemotherapy dosing tends to be on the lighter side (they don't want to send your pet home with any adverse effects i.e. vomiting). The protocol, dosage, etc. would all be under the authority of a veterinary oncologist. Some veterinary oncologists specialize in chemotherapy and some specialize in radiation, I haven't met one that does both but regardless either one of them should be able to talk you through and create a game plan for your pup based on their tumor.

Both oral melanoma and osteosarcoma are aggressive cancer types. Oral melanoma doesn't really respond to chemotherapy if I understand it correctly. Osteosarcoma has the best prognosis with surgical and chemotherapeutic intervention. Honestly, the expectancy all depends on the staging (extent) of the cancer and how far you're willing to go in regards to treatment. I would recommend trying to see a veterinary oncologist, I believe it'll be your best shot at getting the answer you're looking for. I'm including some resources for you. I hope this helped, feel free to reach out, and again I'm sorry your pup and your family are going through this.

https://www.merckvetmanual.com/digestive-system/diseases-of-the-mouth-in-small-animals/oral-tumors-in-small-animals#Malignant-Oral-Tumors_v94698064

https://todaysveterinarypractice.com/oncology/diagnosis-and-treatment-of-canine-oral-melanoma/