Looks like Carolyn Whorton Caladium or miniature elephant ears.

I would re pot it using a good quality potting soil mixture. It should be in a larger pot with drain holes in the bottom and with a saucer so it doesn't hold lots of water in the bottom. All the roots should be in the soil. It sounds like it has very old used up potting mix that it's not benefiting from. When it is like that the water moves through rapidly leaving little behind. Which is why you noticed it feels dry soon after watering. Give it a little plant food every week or so to try to kick start it. It is not a happy plant right now.

Here is a link that might be useful: arrowhead plant

Yes it's a Syngonium - they are a climbing tropical plant and form little rootlets at the nodes sometimes to assist it gripping a climbing surface, or which will form new roots.
Do as ravencajun has suggested and in a while I'm sure you will be rewarded with new shoots and you can dispense with the dead leaves.

I am pleasantly surprised by the quick responses, thank you very much for replying! I am going to try my best to help it out :)

Thank you very much! :)

It could be pruned way back and the cuttings used for many starts. It do root quite well without much help. In our zone some use it for ground cover as it loves to run.

I would repot it in some fresh medium, trimming off any dead or excess roots at the same time. Do NOT place the pot in another container that will hold water....unless you empty it each time.

Do not hesitate to cut out all dead or nearly dead portions of your syngonium. Such pruning.....top and root....is of huge benefit to your plant.

Good news is that this is a very easy plant. You really have to ignore every cultural need for a long time to kill one.

Do a little research and you'll be able to turn this into a lovely plant in a very short time, especially at this time of year.

If you want to post progress pictures, that would be great. We could work through this step by step.

Thank you so much guys! I will be buying some things today and would love to post up some images to show its progress, especially as I am new to all of this. I am actually in Australia though so the weather is becoming a little colder now as winters coming.

Once again, thank you very, very much, I am taking everybody's advice! :)