When I set up a lightning channel with a node, have I need to put some
bitcoins in it?
Yes, you always need to put some BTC, even if the other side want to puts the BTC for you, then that way you cant pay!, you will only be able to receive. To pay you need to put some of your on-chain money in the off-chain contract (LN channel) so that money act as a temporal pool to get from your money in the pool and pay via LN in a fast way!
think the channel as a bag with your money, you must put money in the bag if you later want to pay from there, and you must have room/place in the bag to receive a payment, if not the coins cant keep there
nowadays channels and one side funded, and the initiator is who funds the channel, in the future full-duplex dual funded will be initiated, where both put BTC and it will be send/receive capable.
anyway there are other techniques today to reverse a channel and be able to receive.
Then, if I wanted to buy something using lightning network (for
example, a sticker from Blockstream), will I have to open another
lightning channel with Blockcstream? Or the previously synced channel
will be used to let me buy from Blockstream?
your previously opened (not synced) channel if its connected to a node through which a route to destiny (Blockstream) could be found! will able you to pay, without opening a channel.
if you have to open a channel with everyone you want to pay to. i will be the same as pay with normal BTC, as opening a channel is an on-chain operation. it will only make sense to do repeated payments to the same person, the LN is a lot more useful than that, its great!
think the LN network (when you have a node with good channels opened) as your real life wallet, its handy for small payments, you go and buy a candy so you open your real life pocket wallet and you pay! instantaneously, but you don't carry your hole life savings in your real life pocket wallet!
if you have to buy a house, you go to the bank, ask for opening the safe for you, show your credentials, and you take the money (its more secure but it takes more effort to do) the bank example could be an offline paper wallet of BTC in this example. which if you want to expend from, you have to import the private key in a node, pay and wait for the tx to be included in a block, and wait some more blocks (confirmations) to be safe!... a little more time consuming and impractical than LN don't you think? but.. definitively more secure! like the Safe in the Bank