To give support
help, aid, foster, sanction, approve, incite, instigate, stimulate, goad, spur, sustain, fortify, advocate, reassure, assist, abet, succor, support, befriend, uphold, reinforce, back, bolster, brace, further, promote, boost, forward, favor, strengthen, second, subscribe to, applaud, praise, smile upon, side with, back up, pull for, lead on, egg on, prompt, urge, sway, extend a helping hand, give a hand up, give a foot up, root for, pat on the back*.
Antonyms restrain*, discourage, caution.
To raise the spirits
inspirit, hearten, embolden, animate, cheer, elate, refresh, enliven, exhilarate, inspire, cheer up, brighten, rally, restore, give new promise, give new life, revitalize, reassure, give encouragement, gladden, fortify, revivify, promise well, augur well, buoy up, bid fair, boost, buck up*, give a shot in the arm*.
Antonyms discourage*, depress, dispirit.