I am considering the purchase of two Dittel FSG-5 Airband (118-137 Mhz) transceivers. If I transmit from one radio to the other, and vice versa, am I breaking any rules or potentially causing an issue with pilot/ground crew transmissions?
If you are not using them for aviation purposes or for an emergency, yes, you would be breaking the law, and yes, there is a significant possibility of interfering with aviation-related communications. This will apply in virtually any country.
If you do use them for aviation, and you are a pilot using a radio in an aircraft, then that should be legal. At least in the US, I do not believe any license is required other than your pilot certificate. In other aviation-related contexts, you may be able to obtain a license; I am not familiar with this process, but expect your use of the radios to be strictly regulated, if your license is granted at all.
If you just want personal communications, for non-aviation purposes, a service such as Family Radio Service (in the US) or PMR446 (in Europe and the UK) may be more suitable. Transceivers for these bands should be readily available from sites like Amazon, and possibly from some brick-and-mortar retailers.
Depending on laws in your jurisdiction, and the willingness of you and the other people to whom you want to talk to pay license fees, do some paperwork (or online forms), and possibly take some tests on radio theory and regulations, a licensed service such as General Mobile Radio Service (US-only) or amateur (ham) radio (available in almost all countries, but licensing procedures vary widely) may be an option.
One word of caution regarding amateur radio: many amateur transceivers are available on Amazon without being clearly marked as amateur radios. These include, but are not limited to, "Baofeng," "TYT," and "Radioddity" brand radios. Before purchasing a radio, check the laws in your country and make sure the radio is designed for use within a service that is available in your country, and that you have a license if one is required. Of course, if you have an amateur radio license, it is OK to use amateur transceivers as long as you follow the regulations.
Citizens' Band is available in many countries as well. In the US and many other countries, you do not need a license; in some others, a license is required.