1. Rewrite other people’s short sentences
Take some text with short sentences and rewrite it so it says everything in one sentence.
Children's books are a good source of short sentences. (Well-known children's books are excellent ways to learn a language generally, because they provide the mental "landmarks" that shaped most natives' earliest understanding of the written language.) Here's the first page of Curious George:
One day George saw a man.
He had on a large yellow straw hat.
The man saw George too.
“What a nice little monkey,” he thought.
“I would like to take him home with me.”
He put his hat on the ground
and, of course, George was curious.
He came down from the tree
to look at the large yellow hat.
Rewriting in long, “adult” sentences, here’s what I come up with:
One day, George and a man wearing a large, yellow, straw hat saw each other, George in his tree and the man on the ground. The man, hoping to take George home with him, put his hat on the ground, arousing George's curiosity—an easy thing to do, George being naturally curious—and George came down from the tree to inspect it.
No two people would rewrite it the same way, of course. This suggests another thing you can do: get together with someone else, rewrite the same short text with long sentences, and compare your results. You'll each see how the other person exploits grammatical resources of English that you're not as aware of or accustomed to.
2. Read and write poems
Reading poetry will stretch your mastery of grammar quite a bit. A good one to start with is If— by Rudyard Kipling. It's basically one huge sentence. It's also very well-known: famous poetry, like children's books and nursery rhymes, gives you the same "language landmarks" that native speakers have. It also introduces you to the common wisdom of the culture.
Here's a sample:
If you can make a heap of all your winnings
And risk it on one turn of pitch-and-toss,
And lose, and start again at your beginnings
And never breathe a word about your loss;
It goes on for a long time with many ifs like this before it finally reaches its conclusion in the last two lines of the poem. To understand the sentence fragment above, here are some things that you have to follow: make a heap of means to gather many things into a big pile; it refers to the heap; risk it means to bet it; one turn means a single round of a game; pitch-and-toss is a gambling game; your loss refers to the event of losing the bet or to everything that you lost (it doesn’t matter); and the events follow this sequence: earning a lot of wealth, betting it all at once, losing the bet, starting from poverty again and accumulating new wealth (not from gambling, probably from work)—while never complaining or commiserating. The whole thing is governed by can: the events didn't happen and don't need to happen. It's all just a clause describing the ability to go through all that and not complain; if you have that ability, then…well, you have to keep reading to find out the consequence.
It's okay if you have to read each sentence in a poem many times before you fully understand it. It's poetry. It packs a lot of meaning into relatively few words, and it uses language in extraordinary ways, so it is harder to understand than prose. Native speakers usually need several readings to understand it, too.
But if you really want to master prose, write poetry. This is especially true for mastering the more-complex grammar of long sentences. In poetry, you push grammar to its limits—far beyond the limits of prose. Consequently, when you go back to prose, you find it much easier. You see that prose barely uses the grammatical resources of English at all.
Poetry is hard to write, and it's not for beginners. Even when you have a large-enough vocabulary to think of many ways to say the same thing, you still have to be willing to put up with a lot of frustration as you search for ways to meet the constraints of the form. If you write a few couplets and a few quatrains, you've done well. That alone will increase your command of grammar.
If you take this approach, you must write traditional verse that rhymes and scans, not free verse. Sticking strictly to the form provides the pressure to come up with interesting sentences and forms of expression. Within the limits of the form, you try to be as clear as you can, even as the form pushes you to express things in unusual ways and draw upon unusual syntax. It's not enough to just follow the form, of course; the poem must communicate clearly.