If いきます means 'to go' and いいます means 'to say or call', then how do these verbs convert into て form?
Aren't they the same since they both end in 〜ち、〜り、〜い ＋ ます?
Is this correct? How do you change it?
行きます → 行って → to go
いいます → いって → to say/to call
They are the same, and the forms you wrote are correct, but your reasoning is slightly flawed in the case of 行く.
Godan verbs that end in く usually contract as such:
...and so on. However, 行く is an exception to this (and it's the only one I'm aware of); it contracts as 行く->行って, where you would normally expect it to contract to *行いて. This is just something you have to remember. Your reasoning for 言う is correct, though.
て can do a great many things but is usually not meant to be used just like that. Unless you give an order.
言います －＞ say
言っています －＞ saying
言ってみて －＞ try to say it
言って！ －＞ Say it!
Also, maybe I didn't understand your question well and you were wondering how to differentiate them since they both become 「いって」. To which I would just say to you that in any language it's only natural to have many words written the same way but with different meanings. All you can do is guess with the context.