I'll do a little more research
If you plan to do that research online, you'll probably end up with a pretty fragmented, incomplete understanding, so I'll try explaining relying heavily on the book I recommended.
First of all, like I said, -te iru should be treated separately because its meaning is completely idiomatic and won't help you understand the -te form at all, just like other -te structures such as -te mo, -te aru, -te wa ikenai, etc.
That said, the -te form has a number of uses, some of which seem to get overlooked. Generally speaking, it links two clauses, so when you see a -te at the end of one clause, you know that another will follow. This is possible for multiple clauses, not just two. The meaning, however, varies depending on context, so here's a nice list based on that in my book (C1 can be the first clause, C2 the second). I'll take the example sentences straight out so you can avoid any of my Japanese errors.
(1) Sequence: C2 occurs after C1.
Watashi wa kooto o
nuide, hangaa ni kaketa.
Taking off my coat, I hung it on a hanger.
(2) Multiple states: C1 and C2 are a list of states of someone/something.
Watashi no heya wa
semakute kurai.
My room is small and dark.
(3) Reason/Cause: C1 is the reason/cause of C2.
Kono suupu wa
karakute nomenai.
This soup is spicy and I can't eat it.
(4) Means/Method: C1 expresses how or in what way someone does C2.
Boku wa
aruite kaetta.
I went home on foot (lit. I walked and went home).
(5) Contrast: C1 is contrasted with C2.
Otoko wa soto de
hataraite, onna wa uchi de hataraku.
Men work outside and women work inside.
(6) C2 is unexpected in terms of C1.
Tomu wa itsumo
asonde tesuto ga dekiru.
Tom plays around, yet he does well on tests.
There's still a lot more that could be said about the -te form, which is why I thought it best to avoid explaining here at all, but hopefully this points you in the right direction. The explanations in the book I recommended are more detailed, have more example sentences, formation rules, comparisons with similar expressions, etc.