• Basic forms
• 着る — to wear / base form
• 着ます — polite / I wear / I put on / masu form
• 着た — past / wore / put on / ta form
• 着て — te-form / wear it / put it on (and …) / linking or request
• 着ない — negative / don’t wear / won’t wear / negative form
• 着れば — if I wear / if (someone) wears / conditional
• 着ろ・着てみろ — put it on! try it on! / imperative
• Other common forms
• 着たい — want to wear / 〜たい form
• 着られる — can wear / can put on / potential
• 着られない — can’t wear / potential negative
• 着られた — was worn / was put on / passive / past
• More
• 着よう — let’s wear it / 〜よう form
• 着なかった — past negative / didn’t wear
• 着ている — wearing (right now) / progressive / state

着る (きる) is an ichidan (る-verb). You drop る and add endings like ます, た, て, ない. Used for clothing on the body: シャツを着る (put on / wear a shirt).

• Basic forms
• 着く — to arrive / base form
• 着きます — polite / I arrive / masu form
• 着いた — past / arrived / ta form
• 着いて — te-form / arrive (and …) / linking
• 着かない — negative / don’t arrive / haven’t reached / negative form
• 着けば — if I arrive / if it reaches / conditional
• 着け・早く着け — get there! / imperative (casual)
• Other common forms
• 着きたい — want to arrive / want to get there / 〜たい form
• 着ける — can arrive / potential
• 着けない — can’t arrive / can’t get there / potential negative
• 着かされる — be made to arrive / be forced to get there (unnatural but grammatical) / causative-passive
• More
• 着こう — let’s head there / let’s arrive / 〜よう form
• 着かなかった — past negative / didn’t arrive
• 着いている — has arrived / is already there (result state) / progressive / state

着く (つく) is a godan (う-verb). It’s intransitive — something arrives on its own: 電車が駅に着いた (the train arrived at the station). Conjugation changes the final く: 着きます, 着いた, 着いて.