There are many good and reproducible Application Notes available from Common hplc companies for free underivatized amino acids . The most recent ones use hilic. Depends also on your detector System.
When you decide to use RP, It might be hard to retain the hydrophilic amino acids but good for hydrophobic essential ones.
'Normal phase' had a elution pattern that was similar to GC (~50 years ago) and used the same solvent (like Hexane). It also used a bare silica stationary phase which made it 'hydrophilic' which limited its used since most molecules (like proteins) dissolve in proton donating solvents like water, methanol, acetonitrile...
Thus, it was determined to use these solvents but make the column hydrophobic by chemically attaching octadecylsilane (ODS also known as C18) branches to the bare silica. However, the elution pattern was 'reversed'. Proteins and peptides are analysed by specialty C18 RP columns. Check the websites of Agilent, Waters, Phenomenex, and Restek for amino acids methods.
In reverse phase: you can use water as a mobile phase ( polar mobile phase and nonpolar stationary phase)
Normal phase: nonpolar mobile phase and polar stationary phase
It is really cool when you can use water as a mobile phase as or is cheap, readily available and easy to clean which may not impossible with other solvent systems.
Since amino acids have polar end and non polar end you can use either. But, reverse phase C18 is highly recommended.
In Normal Phase HPLC, the stationary phase is polar ( like Silica column) and mobile phase is non-polar ( like n-Hexane etc) . In Reverse Phase HPLC, the stationary phase is non-polar ( like C8, C18 etc) and mobile phase is polar ( like water, MeOH, ACN etc). For Amino Acid analysis, a better option will be Ion Pair Chromatography technique.
Be sure you know the consequences if you use Ion pairing in your system! You can also look for OPA derivatization of amino acids: good Retention on RP and has a fluorophore. What is your detector?