For example, I know I like Australian shiraz and Californian zinfandel/Italian primativo (same grape). They are medium-full bodied but smooth, and tend to have a wintry or spicy flavor (plum, berries, licorice, pepper, cloves, etc). Although higher tannin levels generally indicates a rougher feel, darker fruits even out the flavor so they tend to be very drinkable.

You could try a wine-tasting class somewhere in your community, to find out for yourself what you like. Otherwise, try a wine shop where the staff is knowledgeable and you can explain to them what you are looking for, they will be able to recommend a starting place. Alternately, you can try a few bottles (by reading the labels and picking one) until you find something you like, and choose other wines of the same grape and similar area.

The cheapest wines will never be the best ones, but you can find decent ones for a reasonable price. I’m not familiar with wine prices in Hungary, but you can get a decent bottle starting around £7-8 in the UK and similar euro prices in other countries.

I could recommend a few, but I don’t know the selection there and our tastes in wine may be very different. Here are a couple I like:
Berton’s Black Shiraz
Ravenwood’s zinfandels (there are several)

While expensive European wines can be very good, I find more good cheap wines from California and Australia. And don’t be fooled by ***** tops, some of the better Australian wines don’t use corks.