Do I spare 80Gb to only keep the blockchain and do I need it to be always present? I am new to Bitcoin, reading a book and just downloaded a client. On site I read the note that I would need about 65-80gb to keep the whole history of transactions.

What happens when it downloads the whole data set? Do I have to keep it on my hard drive all the time I use Bitcoin? Or can I remove it once it syncs or whatever?

The other question is, does it seem realistic to use Bitcoin on mobile devices as the space needed for the blockchain is so huge for mobiles? What are the alternatives to having this data in the internal storage?
There are different ways of interacting with Bitcoin. If you just want to make an occasional payment, and don't want to handle large sums, you'll probably be happy to run a Thin Client on your smartphone. They rely on full nodes to provide data, but when you wait for a few confirmations are sufficiently secure. To learn more about different types of wallets, you may want to have a look at Choose your Wallet.

If you want to support the network or want to make extensive use of Bitcoin for business, you may be interested in running a full node. If you want to have and be able to serve a complete copy of the blockchain, you'll need to keep all of it, but if you're only interested in working off a fully validated representation of the network's state, you can instead opt to run in pruning mode. Nodes in pruning mode are as secure as other full nodes, but cannot serve all historical data.