I'll just come right out and say it: Alicante is not a place I would prioritize going if you have limited time in Spain. Other places like Barcelona, Granada, Toledo, Segovia, and San Sebastian would be a better use of your time here in Spain.
That being said, I still enjoyed my long weekend there.
The city itself doesn't have a whole lot to offer beyond Santa Bárbara Castle, Playa del Prostiguet, and the Explanada de España. I found the city itself to be poorly maintained compared to other places I have been around Spain. Alicante is definitely a beach weekend get away type of place.
To get to Alicante from Madrid I took the Renfe Ave which is Spain's bullet train. This ticket set me back €59 each way. Topping out at 300 km/h (190 mph) we reached Alicante in 2:30.
Alicante can be reached by bus for a lower fare but a longer transit time. You can also access Alicante by bus or train from several other destinations in Spain. Places like Barcelona and Malaga.
What To Do
First and foremost upon arrival you're going to want to pickup a 10 journey TAM (Transporte Alicante Metropolitano) card. It costs just €10 and you can get them at the tobacco stores around the city. If you arrive by Renfe, there is a shop right outside the train station where you can purchase the card. It works for both the buses and the Tram and the 10 journey card is enough for a few days in the city.
Santa Bárbara Castle
The Santa Bárbara Castle is the coolest touristy thing to do in the city. Even better, it's completely free. Just be prepared to do quite a bit of walking. Proper and comfortable walking shoes are a must. To get there, a hike up a mountain is required. If you need to, you can take a few breaks along the way to enjoy the view of the city.
The Castle dates back to 713 when the Moors conquered the city of Alicante. It was captured by the Castilians in 1248 lead by Nicolas Peris. It was assaulted again in a major battle in 1296 by King Jaime II de Aragón. Castilian leader Nicolas Peris told King Jaime II de Aragón that the only way he would take the castle is by personally killing him in the complex. It is believed that the king killed Nicolas Peris at the door to the castle as the Castilians fought to the death.
Between 1701 and 1714, there were several countries trying to take over the country to secede the King of Spain. Holland, the United Kingdom, and the French all occupied Santa Bárbara Castle in this period of time battling each other to gain control of the castle.
It's immediately obvious why this castle was of the utmost importance to so many armies over the last one thousand plus years. Attacking it is clearly a daunting task and the armies that occupy the castle are at a huge advantage over an invading force.
Playa del Prostiguet
Those who travel to Alicante probably come for the beach, and for good reason. It's a beautiful beach and very well maintained. Clearly it's quite popular amongst locals and travelers alike. Just make sure to turn around and enjoy the views of the castle looming over the coastline.
Explanada de España
The Explanada de España is a beautifully tiled coastline walkway dotted with palm trees and great views of the harbor that houses the yachts of possibly some of the richest people in Spain and the region. Along the Explanada de España there is a McDonalds and a Burger King, as well as local restaurants to stop for a bite to eat and people watch. It is also just a two minute walk to Playa del Prostiguet.