tl;dr - if you are going to all the work, and a subpanel, you presumably want a bit more than 20 amps (think it needs to be 30 amps minimum for code these days, and 60 amps is probably better.)
You'll have to dig a ditch. At that point, my opinionated opinion is that you should go ahead and put in conduit, and an additional conduit for any current or future possibility that you might want cable, phone, network, etc out there. Ditches are expensive and a lot of work - conduit is cheap, once you have the ditch...Often cheaper (with wire) than "direct burial" cable, and far more resistant to damage in the future - plus it does offer you the potential of pulling out the wire and pulling in new wire if there ever was a problem - but that's low odds of you making any use of it on he electrical side. Network, quite possible.
Before digging a ditch in a city (especially) backyard, call Dig-Safe and have all (Offically known about) services (gas, phone, electric, water sewer & things you may not know about) located. Turn off the circuit to the garage - I would not worry too much about where it is (unless it runs in a conduit that you might be able to re-use - which is not too likely), but odds are that you'll find it when digging, and it's less exciting if it's turned off when you do.
Any wire used must be rated for wet locations - not difficult, just be sure it is. Any exterior conduit is assumed by code to be wet (and that's generally true.)
If the portion of the backyard you are crossing is not travelled by cars and trucks (not crossing the driveway) depth is sufficient if the TOP of the conduit is 18" below finished grade. Be sure to lay "buried electric line below" tape in the top 6" of the trench fill. If you are not digging below frost line (4 feet or more where you are, probably) you definitely need to bring the ends of the conduit up vertically at buildings, and provide a slip (expansion) joint, as the conduit will move with frost. That's generally needed even if the conduit is buried below frost-line as well, unless it's going straight into a basement below frost-line, but its especially critical when the conduit is above frost line. If you don't find the cost phohibitive, a layer of XPS foam over the top of the conduit provides one more indication that there is something there (when someone else is digging, later) and can reduce frost movement a little bit (or a lot if it's wide.) Alternatively, 2" of concrete over the conduit provides some serious protection on top of the conduit, and reduces the required depth to 6" in Rigid or IMC metallic conduit (which may be well worth it in your situation to save on digging) or 12" in PVC conduit.
If you happen to want a walkway that would happen to run where the electric service would, a 4" thick concrete slab extending 6" beyond the conduit reduces the required burial depth to 4" (ie, right under the slab.)
Look for NEC table 300.5 for more detail.
Run conduit the whole way and use individual XHHW-2 wires
First off, if you're going to run conduit, simply run the entire run in conduit -- it's not all that much more expensive than a cable wiring method parts-wise, and running a fat (say 2") Schedule 80 PVC conduit for the whole run means you can put fatter wires in in the future if called for, as well as making the pull relatively easy for whoever (you or your electrician) does it.
As to what goes in the conduit, once you have a conduit for the whole length of the run, you can run individual wires (presumably XHHW-2 insulated AA-8000 aluminum alloy wires at the sizes you'd be dealing with for 100A) for the hots and neutral, and a relatively thin (8AWG) bare copper ground. This is no more expensive than either a SER or a URD cable in the grand scheme of things, takes up less space in the conduit, and is far easier to pull through the conduit than a fat, stiff cable will be.
Best Answer
"Romex" (brand name) is usually used to refer to NM/B cable, which cannot be run in conduit that runs outdoors. It its not rated for wet locations, and all exterior conduits are wet by definition and in reality.
It's also a huge pain and requires huge conduit to run the UF cable (similar to NM/B, but waterproof, with a gray jacket, typically) in conduit.
Use THWN wires in conduit. If you don't want to run conduit the whole way, use a junction box at the end of the conduit to transition from wires in conduit to cable not in conduit.
Edit: if you want 50A service to the garage, you'll need to dig a new trench for new, larger conduit to the garage, since your conduit is 1/2" PVC. Be aware that your conduit MAY not actually extend the whole way - you may just have two shrt sections where the present cable goes into and comes up from the ground, with the UF cable "direct buried" with no conduit in between. That is a legitimate install method with UF cable (the conduit is "protection from physical damage" at the ends where it comes up) but less desirable (no ability to remove and replace the wire without digging it up - also less protection in the ground than wire in conduit.)