What do the symbols on the map mean while fishing?

Solution 1:

According to the in-game icon reference, the large dots with circles around them represent uncommon fish (red box in image).

This icon key can be accessed from the party menu and pressing the key indicated in the bottom right corner (green box in image) to find the significance of other icons, as well.

Solution 2:

According to this guide on fishing in FFXV, it looks like the bordered circles indicate that the lure you are currently using attracts that fish.

Some fishing holes will have signs posted by them to tell you what kinds of fish there are, but most do not and the signs are incomplete anyways. Circles on the minimap will pulse with a white border if you have the correct bait type to attract that fish (I don’t know if this is skill level related or if it is an automatic feature straight from level 1).

However, this is incorrect, as seen in the in-game reference provided in @Leon7C's answer.

Solution 3:

According to this guide the big or small dots just identify different types of fish. Note: the top comment in the thread has a formatted table for easy reference.

Eg. "Allural Sea Bass" will always be a Large Yellow dot. In general, fish indicated by the large icon (dot with a circle around it) are more challenging than the other fish. Fish indicated by just a dot are typically easier to catch.

The linked guide gives specific lures to use on each fish, but some fish will be attracted by multiple types or simply by any lure. It's a good guide and I've used it quite a bit, because the official guide is severely lacking in fishing details.

Since you mentioned you're just trying to get to level 10 fishing, it would be advantageous to target the large dots - both blue and yellow. They offer more SP (Skill experience points) than the smaller fish, and you'll likely catch a few small fish before the big one takes the bait anyway.