The Bitcoin Blockchain is public. Let's just take Blockchain.info as a wallet just like multibit, electrum etc.

Addresses can be connected to several services depending if the address sends the coins to a known wallet of that service. This can be done by analysing the transactions of that address. For example,
A is a known address of Primedice.com
B is your deposit address to Primedice
C is your Blockchain.info wallet
D is your Coinbase address.

This is how a transaction of yours would look like: D->C->B->A. Although Coinbase can't accurately determine if you used Primedice, they wouldn't want to take any chances and would probably suspend your account if you do this over and over again.

If you really treasure your coinbase account, use a mixer.

To buy Bitcoin from Coinbase, you have to verify your identity with some from of government identification. This contradicts the fact that you said you live a private life. However, there are barely any websites that will allow you to buy Bitcoin without identity verification. If you send Bitcoin to you blockchain wallet and then into a gambling site, then nothing will happen at all. You will be able to use the coins however you want. Although, I don't recommend using Coinbase or blockchain to store your Bitcoin because they aren't very safe. A software wallet like electrum would be better.

This is why we use Mixer services like the one in my signature. You send your coins through these mixer services and it would become very difficult for anyone to follow the coins. < not impossible, but not accessible to the average Joe > They would have to approach the courts to issue a subpoena to get that information.

We do not have to be involved with illegal actions to use these mixer services, but it just adds a extra level of financial privacy that Bitcoin lack as a default feature. < Bitcoin is Pseudo anonymous >

They can determine what address you send to when you fund the poker site if you use the bitcoins that you sent from coinbase to blockchain.info.

Whether or not they can tell if it was a gambling site depends on which site you sent to and how much information about their bitcoin addresses that site makes public or available to coinbase.

Only coinbase employees know if coinbase will try to trace your transaction from blockchain.info, but if they do attempt it, then they will definitely be able to tell what address you sent to. It's really pretty easy to do.

Note that there are steps you can take to make it much more difficult for them to determine what final address you intend the bitcoins to go to, but simply sending to a wallet and then sending from there will not hide where you sent them. It doesn't matter what wallet you use (blockchain.info, Electrum, Multibit, Bitcoin Core, Armory, Mycellium, etc).  The result is the same.  Send to one address, then send from there to another addresss, and it will always be possible for the person that sent to you to know what address you sent to.

I want to make a comment about your post. Coinbase has the reputation of freezing accounts, blockchain.io is more like an HD wallet on the web. You can load your wallet using Electrum or MultibitHD by inputting your secret seed words as far as I know. Please do not lump Coinbase and blockchain.io as the same type of service.

Do you mean that if you buy a bitcoin from coinbase then send it blockchain. If you send the bitcoin to the other bitcoin address, the sender are no longer access the bitcoin that he sent. Only the receiver can access the bitcoin, control it and manage once he received it.

There is no problem with somebody knowing whom you sent your bitcoins if you are not doing any illicit activities then you should not be afraid of something. Anyway the person whom you sent your bitcoins will not be known except if somebody knows your bitcoin address and the owner of the address or the receiver. BUt if no one knows that information then nobody can know whom you sent your bitcoins since what will appear on blockchain is the transaction and the bitcoin address only.

No worries. You are right. Coinbase will have no access to check where you will send your bitcoin once it is already in blockchain walle. They can only find out the one you sent from your coinbase wallet to blockchain wallet. Hope that this will help you. If you still have more questions, don't hesitate to ask. People on this forum are very accommodating

If you want to stay private (or even if you don't, for that matter), do not use coinbase or blockchain for your transactions or storage.
They are virtual banks, they give you an account, but don't give you private keys, they control the coins.
They can impose spend limits, freeze your account, or even confiscate your balance.
They also ask for your ID, so you can't be anonymous, and all your transactions are tied to your name.
Use mobile or desktop wallet for transactions (e.g. Electrum, Mycelium), and paper wallets for storage.
If you don't want Big Brother to trace you, use a mix before sending to your wallet.
If you don't have sole and complete control over the private keys, you don't have bitcoin.

Welcome to the bitcoin world As for your question, coinbase has no connection with blockchain. They are both online wallet provider. Therefore, they cannot trace the bitcoin addresses that you had a transactions with once it came from blockchain already. They will only know the blockchain address where you sent your bitcoin from coinbase.