I live in Russia and bitcoin is prohibited here. Can my internet provider see I'm using bitcoin so I will have problems?
UPD: Right now I'm still syncing bitcoin core. Is syncing as dangerous as participating in transactions?
Yes, they can. It is quite easy to detect an ordinary Bitcoin client; it sends unencrypted messages of a standard and unique format to well-known TCP ports. Moreover, if they monitor all your traffic, it would not be too hard for them to link you with the transactions you send; Bitcoin tries to "cloak" your transactions by sending them out along with other people's relayed transactions. But if they can see all your incoming traffic, then when you send out a transaction that you never received from someone else, they can conclude it is one that you yourself created.
Identifying the recipient of the transaction might be harder, unless it happens to be someone they are already monitoring; then they could see when that person spends the coins you sent.
An encrypted VPN, or proxy service like Tor, can help with this, to some extent. However, then you have to trust the VPN provider not to cooperate with the authorities, or to monitor your traffic themselves and use it for their own evil purposes. In the case of Tor, there might be similar problems if you connect to a Tor node which is compromised; someone who controls all your Internet traffic might be able to arrange for this to happen.
Yes, they can see you. Even if you're using cash in person, you still can be traced. Think about just trying to verify an address in an exchange, because you are interested in that address, there is a chance that you know somebody the address belongs to. When something is prohibited, it's better to work on politics to make it legal then trying to go around it because you will live in fear to be traced one day or an other.
if bitcoin is prohibited there, then your government must has a way to monitor its usage, so you probably could get into problem, but the chance of getting prosecuted is very low because they can't trace who you are paying to, so you can buy drug or child pornography or bombs from anyone, and your government won't know what you are buying, but there is a chance that when the police search your house and find the illegal things that you bought with bitcoin and you will definitely go to jail and get your a$$ cleaned there.