Steam gift process

Gifting games via Steam's store is a complicated affair. As a beginner I'd rather rely on buying your gifts in form of Steam keys.
You have plenty of options where to get these Steam keys. Here you can see some of their current deals.

Gifting through Steam can be difficult. You need to be friends for 3 days and there can't be a big price difference between the two countries.
If you buy a game from another store e.g. Humble Store, Fanatical, GreenManGaming, etc (you can check gg.deals or isthereanydeal to find prices in different stores). From these stores you will get a key you can gift. With this method, you need to check if the game you are gifting has regional restrictions. On some stores, it lists the regions in which the game can be activated.

Thank you both for your advice, this seems much easier.

Currently, buying a gift directly through Steam can only be done by choosing someone on your Friends List and, if they accept, the game is immediately added to their Library. If you gift a game to your other Steam Account it would be used up on that account (so, yeah, don't do that).
Steam also limits which regions you can send gifts to, only allowing to send a gift to friends who don't live in a region in which the game in question would be a certain threshold more expensive to buy.
(There used to be the option of storing gifts in your Steam Inventory and gifting them later but, regrettably, Steam removed that function for newly purchased gifts some years ago.)

Currently, most SteamGifts users primarily use keys purchased from online stores since those can just be send to the giveaway winner's e-mail (which is shown to the giveaway maker). Examples include, but are not limited to, Humble Bundle, Green Man Gaming and Fanatical. Some of the keys there could also have region restrictions, but they are usually listed on the store's page and SteamGifts allows to limit giveaways by those regions for convenience.
So, uh, yeah, I would suggest using keys, since gifting directly through Steam could be rather cumbersome.

You can read more about how gifting through Steam works in their FAQ about Gifting. And you can read more about gifting on SteamGifts in the FAQ in the Help section in the top middle of the page. ^_^

...And, yep, it took me so long to write this that people already answered your question loong ago. XD