To sob means to cry noisily, taking sudden, sharp breaths.
To wail means to cry in a loud high voice.
To whimper means to cry making low, weak noises.
To weep means to cry quietly for a long time.
To bawl means to cry loudly, especially used for young children and for people for whom you do not have much sympathy. As in: “Stop bawling,” Dad said crossly, “and come here right away.”
To cry your eyes out means to cry a lot or for a long time.
To snivel or sniffle is to cry in a weak, complaining way and at the same time breathing in air noisily.
And here are some expressions that you can use in place of ‘cry(ing)‘:
‘be in tears’
‘have tears in (your) eyes’
‘tears run/roll down somebody’s cheeks’
‘burst into tears’
‘eyes welling up’
To mean almost crying, you can use these:
‘be close to tears’
‘on the verge of tears’
‘have a lump in (your) throat’