Depends on your DG; Regulus and UB should be getting items, whereas characters like Sedna and QoT should be more concerned with citadel upgrades and helping out the team. Characters like Erebus and Rook come down somewhere in the middle.
Personally, I think the only citadel upgrade worthwhile at the start of the game is the building hit point upgrade, and in 3v3 or larger I really think someone really should buy it at the start of the game. It allows your buildings to regenerate on their own and is critical for keeping some of those tower lines up into the mid-game. Without it, characters like TB and Regulus can often whittle down their health to the point at which even low-level demigods can simply rush the tower line.
My first item is usually a scaled helm. Boots of speed are wonderful, but if you have to keep going back to town because you're out of mana they're a poor investment. Scaled helm almost doubles the mana regeneration of a low-level character. I typically complement it with one of the low-level armour items to keep me alive. One some maps, stocking up on teleport scrolls is also important; I would never even consider any opener on Crucible where you do not purchase two teleport scrolls immediately. On prison, however, they are not necessary.
I typically will get some basic upgrades early on, such as tower health/damage and currency I, but after that I will focus on items. I keep myself for the most part to the 3000 and under cost range. The more expensive items just aren't worth the expenditure early on compared to some of the more cost-effective stuff in the low-end range. Nimoth's armour, boots of speed, vlemish faceguard, wand of speed; these are the items that usually make up my early game build. The only high-cost item I even consider is the heart of life.
Mid-game, I either save up for some of the high-end citadel upgrades or a high-end item. Regulus or the UB benefit enormously from mageslayer, and this is well worth your time. A tower rook benefits enormously from Heart of Life, also worth your time. For many other characters, including Oak, Regulus, and TB there just aren't great items available for you and you're better off saving for catapults to make a push.
So, as a rule, I keep myself down to low-level items (3000 and under) early on and then save up either for an artifact or citadel upgrades, depending on my demigod. I find this is the best balance since the low-level items are the most cost-efficient and typically citadel upgrades to more to win the game than anything else.
