Yes, I've been to China to train at the Shaolin temple at Songshan mountain in Dengfeng, Henan, China.
I did this only for a couple of weeks, mostly just for fun. My wife (then girlfriend) was training at a school in Houston where the head instructor was an actual Shaolin monk who had moved to the U.S. to start up a school. So the monk offered to arrange everything and take us there with him when he returned. A group of about 8 of us went. We stayed at a house that was right outside the temple grounds, and every day we would get up at the crack of dawn, head down to the temple, train, go have breakfast, train, go have lunch, train, go have dinner, and then head back to our house to go to sleep. And repeat that for 2 weeks.
The first couple days of our trip was actually a small tour of Beijing. That was very cool. We got to see the Great Wall, the Forbidden City, Tienanmen Square, and a bunch of other places. Then we headed by plane to Zhengzhou, Henan. Once there, we got into a private bus arranged for us ahead of time and headed to Dengfeng, Henan. We picked up supplies there, including our monk uniforms, leggings, shoes, toilet paper, water, etc. And then we headed by van to the temple, which is about 15 minutes away, if I recall correctly.
Along the way from Dengfeng to Shaolin Temple, you'll see tons of martial arts schools. One in particular (Shaolin Tagou school) has something like 50,000 students. And you'll see them all practicing along side of the road as you travel on the road to Shaolin. The whole town of Dengfeng is setup to cater to martial artists, by the way. There are martial arts supply stores everywhere. Kung Fu is huge in this town.
So, my advice if you're interested in an authentic Shaolin temple training experience, set it up with a monk or one of his students. There are many monks that offer this to westerners. They will arrange everything: Travel (including planes, buses, vans, private cars, etc.), interpreters, hotel room, meals, travel around the city, your personalized training regimen including your instructors (typically by monks at the temple), equipment and uniform purchases, tuition, tours of Shaolin, etc. They can even have tourist guides sent to meet with you in Beijing, Shanghai, etc. if you want to do some looking around.
I don't think our particular monk would offer to take just anyone. So I won't give you his reference. But while I was at the temple, I ran into a group being lead by Shi De Yang, one of the more well known senior monks there. That group's translator gave me his business card and said that if I knew of anyone looking to train, they should contact him to set it up. So here's his contact. I'm not sure if it's still good:
If that contact is no longer valid, you can probably find out online who's setting these training tours up and arrange things through them. There are actually many monks and many schools doing it. My advice, though, is to make sure that you're training at the Shaolin temple itself, not at one of the neighboring private schools down the road. There's also the International Shaolin Center that's not actually part of the temple but is right next to the temple, and that one seems to be run by the monks with the intention of teaching foreigners. So either the temple itself or the International Shaolin Center next door would be fine. But anything else, you might want to really do your homework before you go.
That's for Shaolin kung-fu training. For Taiji training, you'll probably want to arrange a similar training visit at Chenjiaguo (Chen village). I'm not sure how "full service" those groups are, though. You might have to arrange your own transportation and hotel accommodations. But I bet you can find a group that does all that and gets you in to train.
All of this is possible. It just depends on how long you're willing to stay, and how much you want to pay. But I can tell you, everything in China is much cheaper than most other countries, especially U.S. or Europe. Food, clothing, shoes, etc. are super cheap. You can buy meals for a group of 10 people in China for about the same cost as what you'd pay for one meal in the U.S. Shoes like Feiyue sell for around US $5 there. T-shirts will sell for around US $1. A full monk uniform can be bought for around US $5. You can buy weapons real cheap, too.
So mostly it comes down to how long you're willing to spend there. We met a Russian guy there who spoke English, and he said he was going to stay for as long as it took to lose weight. And he was large. It's not uncommon for westerners to just take a year off and go train.
Just one more thing to note. When you get there, you will get traveler's diarrhea. Almost everyone gets it in China on their first trip. It's just different bacteria than what you're used to. It will last between 2 and 7 days. After that, you're fine. The first 7 days is also what you'll need to get used to the altitude. You'll feel terrible that first week, basically. You'll be tired, nauseated, and anything you eat will give you violent diarrhea. Make sure you have toilet paper and carry it with you everywhere, because they don't provide it in their bathrooms anywhere in China.
Bottom line: Plan for at least a two week trip. The first week you might be unable to train due to these problems. The second week will be much better. Return trips might go better, because your body may be immune to it after that (but don't bet on it).