Can my internet provider see I'm using bitcoin core (Russia) 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.