Engineering is great for creating all sorts of interesting and useful gadgets, but it can get expensive to level up. If you don't already have a huge gold stash consider taking mining to ease the pain of the materials cost.

1-150

1 - 40
Rough Blasting Powder (1 x Rough Stone)
40 - 50
Handful of Copper Bolt (1 x Copper Bar)
50 - 51
Arclight Spanner (6 x Copper Bar)
51 - 65
Copper Tubes (2 x Copper Bar, 1 x Weak Flux)
65 - 75
Rough Boom Sticks (1 x Copper Tube, 1 x Handful of Copper Bolts, 1 x Wooden Stock)
75 - 95
Coarse Blasting Powder (1 x Coarse Stone)
95 - 105
Silver Contacts (1 x Silver Bar)
105 - 120
Bronze Tubes (2 x Bronze Bar, 1 x Weak Flux)
120 - 125
Small Bronze Bombs (1 x Wool Cloth, 1 x Silver Contact, 4 x Coarse Blasting Powder, 2 x Bronze Bar)
125 - 145
Heavy Blasting Powder (1 x Heavy Stone)
145 - 150
Big Bronze Bombs (2 x Heavy Blasting Powder, 3 x Bronze Bar, 1x Silver Contact)

150-225

150 - 175
Blue, Green or Red Fireworks (1 x Heavy Leather, 1 x Heavy Blasting Powder)
175 - 176
Gyromatic Micro-Adjustor (4x Steel Bar)
176 - 190
Solid Blasting Powder (2 x Solid Stone)
190 - 195
Big Iron Bomb (3 x Iron Bar, 3 x Heavy Blasting Powder, 1 x Silver Contact)
195 - 205
Mithril Tubes (3 x Mithril Bar)
205 - 210
Unstable Triggers (1 x Mithril Bar, 1 x Mageweave Cloth, 1 x Solid Blasting Powder)
210 - 225
Hi-Impact Mithril Slugs (1 x Mithril Bar, 1 x Solid Blasting Powder)

225-300

225 - 235
Mithril Casings (3 x Mithril Bar)
235 - 245
Hi-Explosive Bomb (2 x Mithril Casings, 1 x Unstable Trigger, 2 x Solid Blasting Powder)
245 - 250
Mithril Gyro-Shot (2 x Mithril Bar, 2 x Solid Blasting Powder)
250 - 260
Dense Blasting Powder (2 x Dense Stone)
260 - 290
Thorium Widget (3 x Thorium Bar, 1 x Runecloth)
290 - 300
Thorium Shells (2 x Thorium Bar, 1 x Dense Blasting Powder

300-375

300 - 310
310 - 320
Fel Iron Bomb (1 x Fel Iron Casing (3 x Fel Iron Bars), 2 x Handful of Fel Iron Bolts (1 x Fel Iron Bar) 1 x Elemental Blasting Powder(1 x Mote of Fire, 1 x Mote of Earth)) x 10
320 - 335
335 - 350
White Smoke Flare (1 x Netherweave Cloth), 1 x Elemental Blasting Powder (1 x Mote of Fire, 1 x Mote of Earth) x 25
350 - 355
Hardened Adamantite Tube (3 x Hardened Adamantite Bar)
355 - 360
Felsteel Stabilizer (2 x Felsteel Bar) x 5
360 - 375
Khorium Scope (1 x Hardened Adamantite Tube (3 x Hardened Adamantite Bar), 4 x Khorium Bar, 2 x Dawnstone) x 15
360 - 375
Felsteel Boomstick (1 x Hardened Adamantite Tube (3 x Hardened Adamantite Bar), 4 x Felsteel Stabilizer, 4 x Handful of Fel Iron Bolts) x 15
360 - 375
Hyper-Vision Goggles (4 x Heavy Knothide Leather, 2 x Khorium Bar, 2 x Nightseye, 2 x Large Prismatic Shard) x 15

Overall Estimated Required Materials:

55 x Bronze Bars
80 x Coarse Stones
51 x Copper Bars
30 x Dawnstones
50 x Dense Stones
255 x Fel Iron Bars
10 x Felsteel Bars
45 x Hardened Adamantite Bars
50 x Heavy Leather
75 x Heavy Stones
19 x Iron Bars
60 x Khorium Bars
10 x Mageweave Cloth
126 x Mithril Bars
35 x Mote of Earth
35 x Mote of Fire
25 x Netherweave Cloth
90 x Rough Stones
33 x Runecloth
10 x Silver Bars
90 x Solid Stones
4 x Steel Bars
309 x Thorium Bars
29 x Weak Flux
10 x Wooden Stock
10 x Wool Cloth

The above is, obviously, a compact overview of how to increase your Engineering skill to 375. For an in-depth Engineering guide, including gold-making opportunities to be had, please take a look at Penn's WoW Professions guide.