Some key things to look for:
Delivery time. Many low cost fabs require several weeks to schedule and build your boards. Three-day turn-arounds cost much more.
Will they respect your requirements (fab notes), or just build to a standard set of tolerances and specs? Many low cost vendors severely limit what they will accept in fab notes.
Quality. Do you trust this shop to build your board right the first time, or is there a chance they'll mess it up and have to re-do it, causing a delay?
Lower-level materials. Are they buying the actual laminate materials from high quality vendors or just whatever's cheapest at the moment. (Does your application need the higher-quality material?) Will they use the same material for every lot? If they're just buying at lowest cost, the product is likely to vary from lot to lot.
Support. Are they providing engineering support to review your design and catch mistakes (mostly your layout mistakes) before you spend money on the fab.
Test. Are they providing 100% connectivity testing on the boards after manufacturing?
Certifications. Can the provide UL 94V-0 fire resistance certification on your boards?
Capacity. When your prototype works and you're ready to build 1000, 10,000, or 100,000 boards per month, can they support you?
Capability. Do you need 3/3 space/trace, gold plating, thin dielectrics, impedance control, microvias, etc., etc? Higher-technology designs need higher-cost equipment to build and more attention to detail when building them.
Obviously it depends on your project which of these qualities are worth paying extra for.
Should I care the plating? I noticed different processes using tin/gold/Pb and etc.
Traditional tin/lead is easier to work with for hand assembly, but can't be legally used in products you want to sell in Europe, China and probably some other countries.
Pure tin coating is a reasonable alternative to tin/lead if you can't use tin/lead.
Another option is organic solderability preservative (OSP) over bare copper. This is common for volume production.
Gold is useful for corrosion resistance if you are making an edgecard connector, or if you will be wirebonding to a chip-on-board component. Different gold-plating processes are generally used in these two situations. If you need to know the difference, it's worth opening a separate question about it.
If you are hand-assembling your boards and in the USA, tin-lead is probably your best choice.
How could I know the quality of a PCB?
Measuring the quality of PCBs is pretty involved. Most defects happen when you try to push to the limits of technology (3 mil traces, 6 mil vias, 0.5-mm BGA pads, etc). If you are only working at low volumes and with forgiving design rules (8/8 or larger), you aren't likely to run into quality issues, even with low-cost vendors.
Best Answer
What's important to the fabrication shop is not the size of the individual board, but the size of the panel it will be made in. Meaning, typically PCBs are made with multiple images on a panel, and then cut apart either before delivery or after component assembly.
And, yes, there are (semi-)standard panel sizes.
Among those standards, different shops will be set up for different ones. However, every shop I know offers 18 x 24 inches as a standard size.
There are lots of other sizes available, so if 18 x 24 doesn't work for you, contact your shop to see what else they offer.
If the total area of your order (plus some allowances I'll mention below) is less than the smallest panel your vendor can handle, you'll end up paying for some waste. If you try to absolutely maximize the usage of a panel, the vendor is likely to run one extra panel to allow them to still deliver your order if there's a defect in one location, and again you'll pay (a fairly small amount) for this extra panel to be made and then discarded. If you design your board to fit (for example) 12 to a panel, and then only order 10 boards, you're likely to get about the best pricing you can hope for.
When deciding how to panelize your board, you also need to consider the routing allowance needed between the boards and gutter allowance along the edges to allow for handling. In my experience, typical minimums for these are 0.2 and 0.25 inches, but different shops may have different requirements, so confirm this with your shop.
If the boards will be assembled (pick & place'd) in panel form, you should also verify the edge gutter requirement with your assembly shop.