I'm running a character who uses Power Armor exclusively myself, and the biggest piece of advice I can give: figure out how you play. My character only has 6 INT, so I don't have any ranks in Nuclear Physicist, and I'm doing just fine. I'm currently sitting on about 15 Cores, and I've also made it a point to not buy any at all; all the ones I have I've found. I'd be swimming in Cores if I was actually buying them. Oh, and the character is currently Level 21, so I'm not super-high-leveled. The thing is, this works because I don't Sprint, I don't use VATS, and I hardly ever use anything other than a Recon Scope (which means I don't hold my breath a lot to steady my aim), so I'm not putting an excessive drain on my Cores. I can therefore afford to live without the Nuclear Physicist perk. If you're regularly doing the things I'm not, you're probably going to want to invest in the perk.
Keep in mind that there are about 5 Fusion Cores within easy reach of Sanctuary. They are:
1. Under the Red Rocket station.
2. Two at the Robotics Disposal Ground by triggering the self destruct on the Sentry Bot there. You can occasionally get a third one from the military trunk just to the right of the bot, too.
3. Satellite Station Olivia.
4. Concord Civic Access. I'm pretty sure you can get this one even without triggering the Deathclaw; there are several entrances to the Civic Access, and the Core is located in an area that isn't blocked off by the chained door.
And these are all full cores; the half-drained Core you get at the Museum of Freedom isn't included! This is great, as it can allow you to get down to Diamond City much more easily, where Arturo regularly sells Cores. With these to start me off, and heading to Arcjet Systems with Danse, I've literally been wearing Power Armor from the moment I got it. There were a few times I thought I might run out, but I generally found a core or two in time, and once I got to the Prydwyn (I did join the Brotherhood), several more fell into my lap.
Keep in mind that joining the Brotherhood and doing enough to get access to the Prydwyn is the easiest way to get access to a set of T-60 Armor. You can certainly work without it and rely on fixed armor locations to have T-60 pieces, but this is a guaranteed upgrade, so at least keep it in mind.
You're going to want at least 3 Strength and 6 Intelligence. This will allow you to get the Armorer and Science! perks, which will let you build the majority of the mods for Power Armor. The Arm mods will require points in Blacksmith, but it's worth noting that Unarmed damage in Power Armor is lower than out of it, since you can't equip Unarmed weapons. The tradeoff is that the increased defenses make you harder to kill, though, so I wouldn't say Unarmed is gimped in Power Armor; it's just a different focus.
For the purposes of doing the most damage with Melee weapons while using Power Armor, you don't need to worry about your Strength, as getting into Power Armor automatically sets your Strength at 11. This also applies to carry weight. However, the Pain Train perk works only with Power Armor, and requires 10 natural Strength. I do not recommend Pain Train at the start of the game: it requires you to Sprint, which uses AP, which drains Cores faster. It's possible to work around this with ranks in Nuclear Physicist, but setting Strength to 10 and Intelligence to 9 results in only having 4 points left to spend elsewhere, so your character becomes unable to pick up useful perks in other trees without spending several levels investing in the appropriate SPECIAL stat. Better to have a lower Strength stat and build your way up to it later in the game than to screw your build over early on.
Those are the things I can think of off the top of my head. If I think of anything else, I'll post it.