To start a conversation with someone, you could say TheirCall this is MyCall or alternately, MyCall calling TheirCall but the former is more common and slightly more correct.
On HF and weak signal modes, you might say CQ CQ CQ YourCall meaning calling anyone... CQ is repeated several times to give people scanning a chance to catch your transmission, or to give someone time to fine tune the frequency to get you solid, or perhaps turn their directional antenna towards you. By repeating it, you give them time to tune you in before you give your call sign. Otherwise they might get part of your call sign or miss your transmission completely.
Legally, according to FCC, there is no need to begin a conversation with a call sign at all, but this is common on HF frequencies so that the parties involved can know who they are talking to. On VHF where both parties are likely local and can recognize each other by voice, this is frequently still done out of habit, but isn't required and isn't always done.
According to FCC rules, the only "right" way to end a conversation must include your call sign. Anything additional is politeness and informal protocol.
You can just drop your call sign and be done, which makes it legal.
But if you want the other person to be clear of your intent, there's a number of common words and phrases you might add to that.
- Clear indicates that you will be clear of the frequency, probably about to turn your radio off.
- SK is frequently used in morse code similar to clear. This is a Prosign for "silent key" meaning you will stop using the key, you're done.
- QSY means you will move to another frequency, which you likely either mentioned before or will mention now, assuming you have a target rather than just wandering across the band.
- Monitoring or listening indicates you're done in the conversation, but will keep the radio on in the current frequency -- typically used when the other party you were talking to will clear but you will stick around. This might also be used when you put out a call and nobody responded, and you're done calling, but will continue listening. This is perhaps in the vain hope that someone was listening in the background but didn't want to interrupt, but might want to chime in now.
- Clear on your final is used when both parties are done, but you're going to stick around long enough to hear the other person say their final words and clear.
These are all conventions. None are required,