I'm sure you've noticed by now that Mandarin has no 'tense'. Like in English, we say "I wore a red T-shirt"(past tense) and "I am wearing a red T-shirt"(present tense) In Mandarin, there are no verb conjugations. Here's an example: 我穿红汗衫 can be interpreted as "I wear red T-shirts" "I am wearing a red T-shirt" or even "I wore a red T-shirt". Now, if I said: 我昨天穿红汗衫-Yesterday I wore a red T-shirt, it is obvious that the action has happened already. Tense is shown by the time being stated (e.g. yesterday) but the verb does not change. 了 does not relate to past tense, but emphasizes completed actions. Like, in the sentence 我吃了早餐(I had breakfast) there is no clear distinction on WHEN exactly I ate breakfast, but that it has been completed. Which is why it is important to add 了 but lets say your friend asked "What did you do last Friday?" you could answer: 去餐馆吃晚餐-I went to a restaurant and had dinner. There is no 'le' since the time and context is clear.
过 is for positive and negative experience. (Again, there is no clear distinction between when it has happened)
我去过日本东京- I have been to Tokyo.
我没看过这部电视节目 - I haven't seen this t.v. program.
An important thing to note: 过 can be used in positive and negative sentences, whereas in the sentence 他没吃西瓜 "he didn't eat watermelon" 了 cannot be used, but 过 can.
Sometimes people use them in the same way, which can cause confusion. Its better to use 过 when you're talking about experience, i.e. 她去过日本东京三次- she has been to Tokyo three times.
Lets say someone asked: 你功课做了吧?-You haven't finished your homework have you? Then, you could answer: 我做过功课了-I have finished my homework.过 already shows completion, but its also appropriate to use 了 here, since 了 emphasizes the whole action being complete.
There are other words to show completion i.e 到,完,: 找到 -to have found, 完literally: to finish, 我吃完了- I have finished eating. And even 好 as in, 我买好了 - I have already bought[that/it].