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Generally speaking, if a single synapse is changing its likely to be the promotion of more or less receptors for a given neurotransmitter, increasing or decreasing the probability that it will fire in the presence of said transmitter.

Here are some properties which can change at the synapse:

1. The sensitivity of the post-synaptic membrane, making it require more or less input to respond with an action potential at the axon.

2. The amount and type of receptors present in the post-synaptic membrane.

3. Production of neurotransmitters in the pre-synaptic terminal.

4. The branching of axons in the pre-synaptic membrane e.g. after nerve damage you can get axonal sprouting of one axon into a now vacant section of dendrites.

The precise physical basis of memory is still largely unknown.

This is a great answer and I might add an increase in the number of dendrites at the synapse as well. These physical changes are thought to result in electrophysiological changes in the post-synaptic neuron, such as long-term potentiation (LTP). But u/chazwomaq, is correct in that nobody knows how these changes underlie the formation of memories.

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Where does myelination fit in here?

Myelination in human continues up to around 20 years. It improves the speed and efficiency of neurons. Not sure if it is implicated in memory formation, although conditions like multiple sclerosis (loss of myelination) can involve memory loss. Note that myelination is not to do with connections between neurons.

With MS, memory loss is typically more associated with the loss of brain matter (atrophy), rather than demyelination.

If demyelination occurs alone, the memory is typically preserved, it just takes longer to recall. The main component of cognition affected here is processing speed.