Definitely go with grape wine.
Your greatest chance of good first experience with wine is with grapes.
Fruit wines (which is mostly what I make) are more difficult as all the variables have to be dealt with individually ... whereas grapes are closer to the ideal winemaking fruit for all of their profiles ... sugar, acid, tannin, nutrients, they even have yeast “built-in” on the outside of the skins.
If you fell into a vat of grapes and had a seizure ... in a few weeks you'd have wine.
Definitely get at least one carboy.
More than one secondary vessel definitely makes it easier to rack into clean vessels without introducing excess oxygen contact as you would get with a single carboy - which would require racking to a bucket, cleaning your one carboy, then racking back into it.
You could even use gallon glass jugs ... they have their good and bad points. A bad point is that they are not as resistant to temperature change as the much larger carboy. One of the good points is that they are more flexible when you rack. Racking usually results in just slightly less volume of must/wine which means you have to either top off with something (water, juice, marbles, whatever) OR you can rack to a slightly smaller vessel.
Gallon jugs are more flexible in this regard ... and will provide you with a <ahem>, gratuity ... an amount of excess wine that is left over from racking into similar sized vessels ... it is a consumable amount leftover for enjoyment. A slippery slope though ... the habit of leaving a "gratuity".
Alternately to the gallon jugs you could also just go with a full size carboy and use glass marbles for topping off.
As for corks, you might consider trying Zorks. They can be inserted by hand if desired.