These tips are focused on one playstyle more than just general play, but I think it makes the most of their uniqueness.
Leaders
Garrim's good for his Defender, if that trait makes you feel more comfortable, but it really doesn't stand up to Beeri's Industrious. I believe that typically building wonders early on is a bad idea, unless it's central to your strategy or you aren't doing much else. Industrious, however, lets you knock out the Heron Throne, which is a great boost especially with Financial, or the Great Library or others without putting a dent in your early-game production. Either way you're financial, which is what really matters.
As has been pointed out, Beeri is ideal for rushing the Mercurian Gate.
Barnaxus
The manual suggests waiting till you have Blasting Workshops to build him, but I don't find that necessary. You'll have plenty of Fireballs soon enough. I get him STAT-ASAP. If you have a Barrow or Ruins around let him farm that. Once he's up to C5 all your other Golems will have C2.5, effectively, making them 50% stronger. Huge boon.
Gargoyles > Iron Golems
You could use Fireball-throwing Mud Golems if you wanted. Anyway, Gargoyles are best, assuming you have Blasting Workshops, because a)the Machinery tech line is more useful than the melee, b)they have Defender, and c)they build faster with Marble. They make up the vast majority of my military.
If you don't have Blasting Workshops for whatever reason, then you'll benefit from Iron Golems' high strength. Also you'll eventually want Iron for the production, and the top-tier Golems and possibly the Mithril Golem much later on. Still Gargoyles and Clockwork Golems are much more worth the time spent than the melee line.
Advantages/Disadvantages of Golem Race
Golems are immune to Disease, Plague, Death, and Poison. This means that Diseased Corpses and Mary Morbus are great to take along on invasions. It also significantly lessens the threat of the Horsemen and Death summons.
AC 90 & 100 will leave you in the perfect position to wipe out all your opponents' cities.
The major disadvantage that I find is that you can't speed up your golems, as they aren't affected by Haste and can't receive Mobility or Raider. This applies to other magic buffs and promotions, but movement is what I miss most. This can be made up for by the Mounted line, if you can spare the
.
It would be kind of cool if some golems, such as the Clockworks, had a sprint-like spell "Overburn" that gave them +1 this turn and -1 the next. I think it really only fits the Clockworks flavor-wise.
Religions
The best religious path for them is AV followed by ROK. This will make you Neutral and eventually able to make Dwarven Druids. Make sure they are upgraded from Stonewardens so they get Medic and Divine. They are very powerful. The AV path is not out of your way because you'll want Sorcery, and Wicked gives access to Prophecy of Ragnarok. If you're raising the AC then you can choose between the Mithril Golem and the Meshabber, but you really won't
need either.
Otherwise, your choice depends on the situation. You can get Druids with Empyrean, OO, and COE, I find ROK is generally more powerful, and Bambur and Arete help a lot.
Magic
I change Garrim to Arcane personally. I like to use a lot of mages (and their art is some of the best) for Repair, Rust, and Dispel Magic for combat and Inspiration, Hope, and Wall of Stone around the home, with Stoneskin of course and Destroy Undead if the situation calls for it. For the Archmages I use Wither, Spellstaff, and Earth Elementals, changing all my mana to Earth at that point.
Because your golems don't gain experience, the only ways to get Guardsman are with Barnaxus, Bambur/other heroes, and promoting Warriors to Guardsman and then upgrading through to Slingers/Crossbowmen/Arquebusseses. Plan ahead if you're going to use a lot of mages.
Armageddon
You'll notice I focus on a high AC. This is ideal because the effects are mitigated for you, and you can build the Mithril Golem. All you need for this is the Prophecy of Ragnarok. Although it doesn't affect your golems, you should spam Adepts in this city, as it's useful to have 2 or 3 Repairers in each Golem stack. Diseased Corpses are another option if you stay AV.
If you're not going to superexpand and conquer lands, then there's no need to raise the AC.
Techs
Your priorities after your initial improvement, economy, and religion (as appropriate) techs are Construction for Barnaxus and Wood Golems, Sorcery for Blasting Workshops and Alchemy Lab (Mary), and the Engineering/Machinery line.
Civics
The lack of experience for golems reduces the benefit of Theology, Apprenticeship, and Conquest. You're Financial so Agrarocracy or whatever you wanna call it is powerful. I usually go with Arete and Military state as well.
Worldspell
I find that the benefits are less noticeable later on, so I use it after I've settled my first ring of cities (3 or 4 or so). What to do with the Hammers? I usually split them between two cities because I don't like to completely choke my GP production; I'll put at least three in the capital and the rest in my designated GP city. I might give one to Barnaxus, but only if he's really working hard. Even earlier timing of this spell can help with rushing military or wonders.