Tip #1: You cannot just show up on any day of the year. They no longer sell entry tickets at the park. You must arrive with a ticket in your hand.
Tip #2: The tickets now include a shuttle bus from one of the Metro stations, so that will be the easiest/cheapest way to get there.
Early morning is a good idea during the hot months, for sure. I am cold-natured and it would not be my choice in March. I'd risk crowds to avoid (potentially) shivering. It's an outdoor site, so you'll still be able to see the Gaudi architecture even if you're unlucky to be there at a busy time. Other people will certainly disagree with me on this--possibly also you! If you prioritize trying to make photos with no people in them (good luck with that in Barcelona), you should go early.
If you want to see the interior of the little gatekeeper's house (which no guidebook I've seen has deemed essential), you should try for the earliest entry time, line up before the park opens and go straight to the cottage. There's a separate line for it, and that line will quickly build up until you're looking at a long wait. I had the second entry-time slot, and by the time I noticed the line for the cottage it was 45 minutes long. It wasn't important enough to me to stand in the line, so I can't tell you what you would see inside the cottage.
I also have not been to the free-admission part of the park.
I don't think a tour is necessary. To me it's a straightforward visual experience; it's not like a church, where you may need someone to explain a lot of symbolism.
Edited to add: I agree with Carlos that if your time is tight, Parc Guell can be skipped. I'd say the same thing if you're unlucky with the weather. The park wouldn't be so much fun in the rain (which it was what it was doing at the time of my first visit on a chilly, wet day in May 1980ish--pre Global Warming).