How do I reconnect a garage to a house with existing underground wires?
You can reuse the wires one of several ways.   

Run a separate ground wire

Since it is a retrofit, you can add ground wire.  (NEC 240.130).  The 3 circuits can share one ground wire if they all originate in the same panel (as of NEC 2014‘s new 240.130D). The ground wire must be installed properly, and an appropriate size for each circuit it's grounding.  If your fattest conductor is 10 AWG, a 10 AWG ground wire can protect all 3 circuits under NEC 2014.  If I recall, 8 AWG bare copper wire is readily available.

Hots and neutrals from the same circuit must be routed together.  A 240.130 retrofit ground may follow a different route.

If an existing wire is already bare and appropriate size, you already have your ground wire!  

Wire larger than 6 AWG can be redesignated as ground by wrapping it in green tape at each accessible point.

Ask your inspector to re-designate

Generally you cannot re-designate a conductor to be ground, although an entirely bare copper wire can only be a ground.  

Ask your local inspector if there's a way to allow this anyway.  Local jurisdictions can override Code.

Forget ground and use GFCI

GFCI's may be required anyway for your new circuits.

You can use the GFCI rules, which allow ungrounded 3-prong outlets when fed by a GFCI breaker or outlet.   All the outlets must be labeled "No Equipment Ground". 

They must also be labeled "GFCI protected" if they are a plain outlet fed by a GFCI upstream.

In this case, take care not to connect ground to anything.  Nothing is safe except a ground wire run all the way back to the main service panel.   

Make the barn a main panel, isolated by a transformer

This is a complicated one.  Wire the barn as a main panel.   It must have its own ground rods and neutral must be bonded to ground in the panel.   However this neutral must not bond to the neutral back at the house.
It is isolated from the house by a transformer.  This is a "mini" version of the service the power company provides to you.

The key is a transformer of appropriate size.  It must be single-phase and with a VA rating exceeding your planned load. I often see them on Craigslist at sane prices, e.g. 5K VA for $100.  Typically, you can jumper the secondary to either supply 240/120 split phase like your house, or 120-only at twice the amperage. 

For amp capacity of a transformer that you've jumped for 240V, easy peasy - divide VA by 240. For instance, A 5K VA transformer is 5000/240 or 20.83 amps.  It can be supplied by a 20A breaker and 12 AWG wire, and can output 20A to each side of the panel.  If jumpered for 120V-only, it can provide twice that or 40A.  

Main breakers that size will be tough to come by, don't bother, just backfeed a regular breaker. 

Multiple circuits? Somewhat.

Generally NEC allows only one circuit per voltage. So you can have 240V on one pair of wires, and 120V on a second pair.

You can have more circuits for specialized uses which require a separate circuit, for instance the third pair of wires can be used for lighting controlled from the house. I would use that loophole to power the barn interior lighting on a separate circuit from the outlets - that way if the table saw kicks and trips the breaker, you aren't plunged into darkness with your hands 3 inches from a spinning blade.

Re-designate wire? Mostly no.

Generally, wire function is decided by color, and they cannot be re-designated unless they are larger than 6 AWG (at the request of distributors, who do not want to stock multiple wire colors in large sizes).   However, neutral (white or gray) wire can be re-designated hot (e.g. to allow 10/2 to feed 240V loads).  Also, an entirely bare wire can only be used as ground.