If you want to be anonymous, then there are two aspects to consider, IP address leakage, and address tracking. Both are a concern only when they reveal payments that can be tracked to you, since Bitcoin itself isn't illegal anywhere (although your use of Bitcoin could build a case: "We intercepted drugs mailed to him,
and we know his IP address runs Bitcoin, a nefarious underworld currency only used for buying drugs".)
You can install the software, and synchronize the blockchain without revealing any information about yourself, except that other Bitcoin users may be able to detect that you are also running Bitcoin at your IP address. There is one exception to this statement, a
recent disclosure documented a potential issue where an attacker might be able to discover if a particular address is owned by a particular peer.
When you receive money, no information is communicated with your client specifically, the payment is just recorded in the blockchain, a payment that only you can spend.
When you send a payment from your own client, however, the transaction is broadcast on the p2p network to all other clients to which you are directly connected. There is a relatively high chance that another well-connected Bitcoin client can determine the IP address that initially broadcast the transaction if mitigation is not done (such as limiting your Bitcoin to only one edge-of-network peer connection to send your payment.)
If you are in a hostile regime where all Internet activity is recorded (perhaps the USA, although the drone strike from above would be the only indication that your funding of terrorism was detected) then TOR with a foreign exit node and a state that doesn't have the resources to get a majority of TOR nodes to corroborate ins and outs would be necessary to anonymize your connection and payments.
The second issue is your conversion of goods and currency to Bitcoin, the trail of addresses is usually easier to follow than IP connections, if specific anonymizing steps are not taken. If I buy or sell something from you, I might send you a payment that is later mixed or connected with other payments you have made or received.
