I don't know anything about Adobe Audition, except that you are unlikely to find it in any "Top 10 DAWs List" on musician forum/web sites.
Audacity is a useful tool to have for odd audio-related jobs or podcast recording, but could hardly be described as a DAW.
A decent Audio interface will include some form of direct monitoring, which will reduce latency (in some cases, it's described as zero-latency monitoring). The important thing is that the signal goes directly from the input to the monitor output of the interface (via a built-in mixer that mixes it with the output of any existing tracks), rather than going to the computer, being processed and mixed by the DAW and then returned to the output.
Personally, I use Reaper as my DAW. The home use and professional versions are identical (they just trust you to pay for the professional licence if using it commercially).
Having said that, most Audio interfaces come with a bundled "lite" version of one or more well-known DAW packages.
Another advantage of using a dedicated interface is that you get proper connectors with correct impedance matching for mics (XLR) and instruments (Jacks). Many also include switchable phantom power on the Mic inputs so that you can use a condenser mic.
Focus right are a very popular brand, and they make interfaces for both home and professional studios.
The Focusright Scarlett Solo (their cheapest model) comes with Mic and Line/Instrument inputs, phantom power, and direct monitoring. It is bundled with Pro-Tools First ( a "beginners" version of Pro-Tools, which is probably the most well-known professional software used in recording studios), and Ableton Live Lite).
This is the Scarlett Solo (scroll down for a video which directly addresses questions of latency)
https://uk.focusrite.com/usb-audio-interfaces/scarlett-solo
It's about 95 quid from Thomann:
https://www.thomann.de/gb/focusrite_scarlett_solo_2nd_gen.htm