By other symbols of the Monarchy I am thinking the crowns, the Royal Standard and other such items. I would assume people didn't see the need to keep them as they didn't forsee the monarchy returning. So were they kept somewhere safe? Or am I totally wrong and were they destroyed and Charles II had to have a new set made?
Let's discuss two sets of Crown Jewels, both of which were already in existence during the reign of Oliver Cromwell: the English Crown Jewels and the Honours of Scotland
English Collection
You are correct in your assumption that most of them were destroyed - at least referring to the English collection. The most important part of the collection at that time was St. Edward's Crown, which was used to crown the sovereign at the coronation itself. It was also a holy relic in itself, alleged to have come from St. Edward the Confessor himself. The consensus on the historicity of this claim is doubtful, but nonetheless, the name has stuck. During the time of Cromwell, the crown, along with various other regalia (the scepter, the orb, etc) were either sold or melted down, according to the Royal Collection's site.
The oldest extant piece of the English collection that dates back pre-civil war is the anointing spoon. It's very survival was a matter of luck. It was purchased by a certain Mr Kynnersley, Yeoman of Charles I's Wardrobe, who subsequently returned it to Charles II when the monarchy was re-established. We can that this piece survived not because they were kept safe, but out of sheer luck. In other words, preserving the collection was not a priority for Cromwell and his government. Here is source on the history of the spoon
Scotland
The Scottish Crown Jewels (as they are informally known), are actually older than the English Crown Jewels. This set of jewels were kept safe during the Cromwellian era. Cromwell was well-aware of the existence of this set of jewelry and sought to have them destroyed, but they were kept hidden in various locations, such as Dunnotar Castle and under the floor of Kinneff Parish Church. Although these jewels survived this period, they have never been used again in a coronation ceremony. Source
When Charles II was crowned, St. Edward's Crown was remade, although it wasn't an exact copy of the older, lost crown. Still, this is the same crown used to crown British Sovereigns until now. Source
So the English Crown Jewels didn't survive - except for some select pieces, but most of the Scottish Crown Jewels were kept intact and safe.