Saving Western Rome

[Winterstürme]

How the heck are we supposed to do this?

[BatGoat]

Barbarian Invasion? There's a really good guide at http://rtw.heavengames.com/

[econ21]

IIRC, it's all about turn 1. You have to deal with the insolvency and instability of the Empire, before you have a solid foundation to fight off the hordes.

(1) Disband most of your units - primarily your foederati (which are close to worthless) and your navies - to reduce your upkeep. You need a few frontier armies to go head to head with the hordes, but that's it. You want a decent net income per turn to allow you to invest.
(2) Demolish unneeded buildings for some cash - I went pagan, so demolished churches in most of the empire. I centralised military recruitment in Italy, so demolished military buildings elsewhere.
(3) For public order, religious division is a big deal so make each settlement go one way or the other. African and Italy are naturally Christian, elsewhere naturally pagan. Demolishing the "wrong" religious buildings and building the "right" ones can be quick ways to stabilise public order. Overtime, you can make all your Empire one religion or another, but it's not necessary and in the short term, avoiding going against the grain is the best.
(4) Micro-manage taxes - drive them down if you need to avoid revolt; push them up where you can and don't need a settlement to grow (you might want growth for protective walls or a higher tier military recruitment building).
(5) Move your capital to Massila to reduce maintenance costs.
(6) For your field armies, perhaps the most important addition are archers - they really help against hordes and are low cost; having 4 or so would be good. Later on, you want plumbatari and Sarmatian cavalry. Phase out limitaeni asap (and foederati on turn 1).
(7) Where possible fight on bridges as chokepoints. Walled settlements are also great, but do trap the defenders (hordes can use multiple stacks in a siege so be careful).
(8) Develop night fighting generals so you can salami slice the horde stacks. Learn to let them go when their loyalty becomes too low.
(9) Take out settled minor enemies on the frontier - the Alemanni, the Picts, the North Africans - so you can focus on the hordes. Try to avoid war with Eastern Rome if you can; keeping a decent garrison on the border may help.
(10) Invest money in things that raise your province income - the details on each settlement will show you how much income will rise once a building is completed. Often ports are good investments, as are mines. Returns tend to diminish, as the cost of each level of building rises steeply. But your Empire is so large, quickly you can become an economic superpower.

Interestingly, a lot of the same points apply to the Western Rome campaign in Attila, which you might want to try if you like Barbarian Invasion. WRE in Attila is my favourite Total War single player experience to date.