Naming functional groups in a molecule
First molecule:
The ester group is correct - you have a carbonyl (C=O) carbon bonded to an oxygen and both sides are bonded to an R group (any carbon/hydrogen group).

Phenol is incorrect - for a phenol group, the hydroxy/OH must be directly bonded to the ring, like so:



So instead, what you've labelled comprises of two functional groups: the ring is an arene group, and the OH is a hydroxy/alchohol group.

You haven't labelled the two branch alkyl group on the left of the molecule - this is called an isopropyl group (it has three carbons, so it's basically a propyl group with something attached to the middle carbon). This group might not always be considered a functional group though (maybe ask your lecturer?). Or it can just be called an alkane group.

Second molecule:
The amide group is pretty much correct, except it should only include at most the atoms bonded directly to the nitrogen and carbonyl carbon.

The alkene should only include the double bond (while the whole chain could be considered an alkene, the alkene functional group only contains the double bond).

One that you haven't included is the ether group, R-O-R.

To learn these I would suggest just using a table of common functional groups until you start to remember them. I think the Wikipedia page is good because you can look at your molecule and see what unusual atoms you have (e.g. nitrogen, oxygen) and see if any of the groups under that heading look like they're in your molecule, except it might be missing a few e.g. phenol.