Never served and I'm British so take both those points in mind when I comment..
1- Infantry will be what you make of it - there are some scarily smart people in all trades - and if comments from anyone I know with UK military experience are anything to go by, you will have opportunities for distance learning during deployments. You could pick up a bunch of certifications and credits - and infantry (as with any physically demanding role) has a shelf life - depending on where and how you serve, by the age of forty, you may have knee caps that would look better on a sixty year old, some permanent back or neck pain. I'm not trying to put you off, I'm just saying..humping a pack and a rifle works better in your twenties and it's best to be realistic and plan for what happens later in your life. So, save some money, get some quals.
2 - Not true - the military (quite correctly) takes a fairly different view about even junior leadership and if as an employer I were looking at a CV with any evidence of progression in rank my ears would be pricking up right away. In civilian life, team leader jobs get handed out to whoever is standing around and the results are often ugly. I like working with ex forces people because (as I put in indelicately to a colleague) they've had a decade of being told to wind their neck in and get on with it. They're usually solid performers with tolerable social skills, self motivated and are often solidly reliable team members.
Against that? Pick a trade and you could be making a lot more money both in and out of the forces, plus your knees and back might hurt a bit less in the mornings when you're forty.Not always true - friend of mine got tapped as a clerk in the Royal Marines, developed an interest in IT and is a contractor on about twice what I earn with a degree so..
Up to you, but best of luck