Goa is well-connected to major cities in India, and a few cities in the Middle-East, via the Dabolim Airport (GOI).
For the adventurous, there are several trains from Mumbai to Goa. The "Tejas Express" (22119) is recommended. To reserve a ticket via the Indian Railways website, you may need to create an account first. Please find further instructions here.
There is also a reasonably affordable (about $100) cruise ship option now! Please note that, given the weekly schedule of the ship, this will require you to leave from Mumbai on Friday evening and arrive on Saturday at Mormugao (about an hour by road from Panaji).
If your hotel has a pick-up facility, please contact them to arrange pick-up from the airport. Please note that the hotels may offer pick-ups only at specific times of the day.
Otherwise, from the airport (or the railway station), one can hire pre-paid taxis which can be booked at designated counters near the exit. You may need to make the payment in cash. Please be aware that if you are hiring a taxi from elsewhere, you may be over-charged. So you may want to agree on a fare before starting the trip. The fares vary quite a bit across different kinds of taxis (the white "tourist taxis" being more expensive than the "yellow and black taxis").
Cidade de Goa is about 30 km from the Dabolim Airport and 20 km from the Karmali Railway Station.
If you are not in a hurry and do not mind the hassle, privately operated buses may work for you. Public transportation buses are also available, but this involves first going to the nearby town of Vasco da Gama before catching a direct shuttle to Panaji.
The adventurous can rent a scooter or car for INR 300-400 per day. Your hotel may offer a rental facility. Do remember to carry your driving licence.
Goa also has "motorcycle pilots", which may be an option if you are travelling solo and very light.