More information is needed. US paper currency has a lot of letters in different places that represent plate numbers, position indicators, and so on so simply saying it has a "G" on it isn't specific enough. In addition, the 1934 series of silver certificates included $1, $5, and $10 bills AND there were several sub-issues of $5 and $10 bills.
Please post a new, separate question with the bill's denomination and what letter if any is near the date (e.g. 1934, 1934 A, etc.) You don't need to include the serial number, though.
Normally it would be necessary to have its denomination but the only bill fitting that description is a $1 silver certificate. Please see the question "What is the value of a 1935 G US 1 dollar silver certificate?" for detailed information.
Please check your bill again. "E" is the highest series letter for 1928 $1 bills. 1935 is the only date that extended to G, and these have the standard Great Seal reverse, not the much-scorned "funny back" design that lasted from 1928 to 1934.
1934 $10 silver certificates only carried series letters A - D. Please check again and post a new question. You may have confused the series letter with part of the serial number or a printing plate id. The series letter will be right next to the date; note however that the first issue in a given year is always blank. That means an "A" bill is actually the second series, "B" is the third, etc.
"E" is the highest series letter for 1928 US 1 dollar silver certificates. The only $1 bills with a G series letter were dated 1935. Please check your bill again and post a new, separate question.
There is no "G silver dollar" (a coin) so I assume you're referring to a silver certificate. Please see the Related Question for more information.
B is the highest series letter on a 1957 $1 silver certificate, so you may be looking at a plate indicator or some other marking. In any case 1957 $1 SC's are not rare; in average condition they generally retail in the $1.25 to $2.00 range regardless of series letter.
Normally it's not possible to estimate a bill's value without a date and denomination, but the only silver certificates to carry a "G" series letter were 1935 $1 bills. Please see the question "What is the value of a 1935 G US 1 dollar silver certificate?" for more information.
There was never a 1934 G US $10 bill. The highest series letter, which appears by the date, was D. The "G" is almost certainly the letter code for the Chicago Federal Reserve District that distributed the bill; that's not the same as the series letter. Check to see what letter is next to the date. It could be blank or A to D. Then see "What is the value of a 1934 [letter] US 10 dollar bill?" for more information.
The banner across its top and the blue seal indicate it's a silver certificate, a form of paper money that was discontinued in the 1960s. Please see the question "What is the value of a 1935 G US 1 dollar silver certificate?" for detailed information.
All 1934, 1935, and 1957-series $1 silver certificates had the same color scheme: serial numbers and the seal were in blue while the rest of the bill's front was black and white. The reverse side was almost identical to today's $1 bills, in green and white.
The motto "In God We Trust" also appeared on 1935-series $1 bills with an "H" series letter as well as "G". Please see the question "What is the value of a 1935 H US 1 dollar silver certificate?" for more information.
The face value of any bill or coin simply refers to its denomination, so the face value of any $1 bill is one dollar (and the face value of a quarter is 25¢, of a penny is 1¢, and so on.) The collector or market value of a bill or coin is what it's actually worth to a collector on the open market.However, there are no 1934 silver certificates matching the series 1934 G. Both $5 and $10 silver certificates only extend to series 1934-D. A "G" might appear on a Federal Reserve Note, but (a) those bills aren't the same as silver certificates and (b) the letter indicates the Federal Reserve District that distributed it, not the series letter.Please check your bill again and post a new, separate question with its denomination, seal color, and what letter if any is next to the date.MoreIn the case of a rare item the market value is almost always higher than the face value. Common items - e.g. a circulated 2010 dime - have no extra market value so their face value is the same as their value to a collector. And in some cases, for example obsolete currency that's not in high demand, the face value can be more than the collector value.