Good advice above!
Nothing says you can't manage all this manually, but sooner or later, a mistake is likely to happen. Most people choose to have either dedicated outlets, or a built in transfer switch.
You must do some planning. It is necessary because of the limitations of batteries and inverters. List the 120 volt AC devices you wish to run and for how long you expect it will run each battery charge cycle.
We can then give more specific advice.
The proposed system must consider how to manage the built in battery charger as mentioned above.
The battery bank needs to be large enough to support peak power draw and large enough to sustain power delivery through a battery charge/discharge cycle.
Strategies that tend to work:
A small inverter system may have its own dedicated electric outlets. Plug in to dedicated outlet to run an appliance on the inverter output. Plug in to existing outlets when on shore power.
I have a 1000 watt inverter I run in this way. I charge computer and phone batteries and have on occasion run a CPAP machine.
To use the existing electric outlets in the RV, it would probably need a 2000 watt inverter or bigger. A minimum 400 amp hour lead acid battery bank would be required to support a 2000 watt inverter.
The built in battery charger must be turned "off" when inverter is in use. Air conditioning or electric heaters would not be usable using such a small system.
This tends to work better if the inverter has a built in "transfer switch". It would switch from shore power to inverter and disconnect the built in charger automatically. It also needs to "bond" the system ground and the neutral wire from the inverter. It would remove the bond for shore power use.
A bigger battery bank and inverter would be needed to supply all RV 120 volt appliances as they are normally used. Running high power consumption appliances like air conditioning would require an obscenely large battery bank.
I wish you good luck and happy trails ahead!