Be aware that lo-sec rats may be a challenge in a Destroyer or Frigate, and that people will be hunting you. If someone warps in while you are clearing rats, assume they will attack you first. You’ll be a nice easy target for most of the locals and since ratting locations appear on the probe scanner and you appear on the d-scan they know you are there and what you are in and thus know you are a good (killable) target.
Otherwise, go for it!
(Just want you to be aware of the risks - don’t stay near the warp in point, use your d-scan, if something approaches you don’t like then run)
Ammunition types: highly dependent on your preference for close in brawling or kiting. Both ships are fitted with close range weapons, so go for high damage. https://wiki.eveuniversity.org/Turrets is a good starting read and gives the ammunition types. In Amarr Lo-sec you’ll want to be doing EM/Thermal damage against blood raiders. https://wiki.eveuniversity.org/NPC_damage_types is worth bookmarking.
Seriously consider faction (Navy) ammunition - such as Imperial Navy Multifrequency crystals. The damage done is a lot higher (20% as I recall) and so worth the cost. You didn’t state if you are using T2 guns - if you are then take the advanced ammunition as well - use the short range high damage on things bigger than you.
I suggest you use the usual like Void S (short range, high damage, okay tracking), Null S (longer range, okay tracking) and Antimatter S (short range, high tracking)
That depends on your ranges to the rats, so Scorch S, Conflagration S (short range, high damage), Multifrequency S (short range, high damage but much better tracking), Standard (least capacitor usage, close to Scorch S range but better tracking) and Radio S (long range, high cap usage, good tracking)
And lazor crystals switch instantly, so don’t worry about switching too often.
Your weapons for both ships will give you a choice for short range, high damage or long range low damage. Both are useful. I fly a Retribution with Small Focused Beam Laser II. With Gleam I get 325 dps at 6Km optimal. With Aurora, dps drops to 186 but optimal range increases to 44Km. The ability to quickly switch between them is powerful.
I say fly the executioner and use imperial navy multifrequency S. Get up close and personal. Carry a set of standard multifrquency crystals as a backup.
It helps if you scream motivational battlecries while charging / fighting. Things along the lines of: “EAT LAZER PUTRID HEATHEN!” “OH, YOU WANT SOME?! YOU WANT SOME TOO?!”, etc seem to increase general dakka.*
Lazors serve pilots who tend to tracking / transversal well.