For strictly a list of inequality metrics, see Common Income Inequality Metrics. This article section lists the Gini Index, 20:20 Ratio, Palma Ratio, Hoover Index, Coefficient of variation (of income), Wage Share, and Theil Index.
One of the best all-in-one measures currently available for national prosperity and quality of life / general welfare is the World Happiness Report.
Data is collected from people in over 150 countries. Each variable measured reveals a populated-weighted average score on a scale running from 0 to 10 that is tracked over time and compared against other countries. These variables currently include: real GDP per capita, social support, healthy life expectancy, freedom to make life choices, generosity, and perceptions of corruption.
As you can see, it does measure various indicators of economics, quality of life, well-being, and freedom. It also has GDP per capita built into it, which is an indicator of average wealth and economic power (adjusted for population size). It is not perfect, and I would prefer something that incorporates public debt as a percent of GDP, Gini Index, Unemployment Rate, and some measure of poverty rate (it can be hard to decide relative poverty rate vs absolute poverty rate).
(A lot of derived opinion follows. Most of it is clearly intuitive, but I will try to come back and find sources for claims that seem like leaps and require proof.)
Public debt as a % of GDP is a worthwhile economic stat because it appropriately punishes running-up debts and rewards paying off debts, like you would want. But it also rewards budget deficits that are justified by virtue of growing the economy better than the magnitude of the deficit. Similarly, it punishes repayment of the debt if it is done in a way that grossly sabotages the economy. Another virtue of the statistic is that it is not overwhelmingly biased by size of population, since public debt (numerator) and GDP (denominator) are both proportional to size of population, making the ratio population-agnostic; a statistic like absolute public debt would be flawed statistic for national comparisons because it would reflect larger magnitudes for countries with larger populations. Public Debt as % of GDP is also not overwhelmingly biased by inflation, since public debt (numerator) and GDP (denominator) are both proportional to inflation. Public debt per capita would be a flawed statistic for comparing national economy across various eras, because, for instance, America's public debt in 2018 and America's public debt in 1950 would not be a like-for-like comparison if you don't adjust for inflation.
Gini Index, or some other inequality statistic, is important to take into account because a powerful national economy does not necessarily imply prosperity for all citizens. It could be that all wealth is controlled by a few aristocrats, while the remainder of the population is severely poor. You can't effectively know whether a high GDP per capita implies good prosperity for all citizens if you don't account for the distribution of wealth as well. Therefore, something like Gini Index, Median Income, or Poverty Rate must be used. But Gini Index gives you more texture than Median Income, because 48% of the nation could be living in deep poverty, and as long as the 50th percentile of the population makes good income, Median Income would imply that all is well.
Unemployment Rate can be both a useful and misleading indicator of general welfare. In my observation, some of the most authoritarian, nationalistic countries can run at near 0% unemployment (presumably through use of force to compel their citizens to work), and others may not report on their unemployment rate. Some examples to support this claim: Cuba's unemployment rate is 2.6%, Vietnam's is 2.2%. Meanwhile, there are some Social Democracies with very high quality of life that have extremely generous unemployment insurance and run at relatively higher unemployment rates, affording their citizens better opportunity to pursue arts and hobbies instead of chasing profit (Sweden and Finland seem to be decent examples, with unemployment rates of 6.2% and 7.6%). However, especially in more Capitalist countries with weaker social safety nets, high unemployment is considered a potential predictor of economic recession, poverty, and psychiatric depression. Although poverty rate is probably a more direct and accurate way to measure poverty, unemployment rate is serviceable as another indicator of how many citizens are being left out in the cold by an economy that does not provide enough opportunities for everybody.
Poverty Rate is one of the most informative measures of general welfare, since it directly measures how many people live in poverty. Unlike some other measures of general welfare (e.g. the World Happiness Report), this measure is directly related to poor quality of life that is caused by a weak economy. A problem with the absolute poverty rate is that, within the perspective of developed countries, a lot of the population that we consider poverty-stricken lives well above the absolute poverty line, and therefore will not show-up when you're trying to accurately compare poverty in two developed countries. The problem with relative poverty rate is that, because the standard varies by country, the stat measures countries against an inconsistent standard.
Definitions of the poverty line vary considerably among nations. For example, rich nations generally employ more generous standards of poverty than poor nations. Even among rich nations, the standards differ greatly. Thus, the numbers are not comparable among countries. Even when nations do use the same method, some issues may remain.
As an illustration of these different measures in action, note that 51.4% of Turkmenistan lives in absolute poverty by the World Bank's definition of $1.90/day, whereas 0.0% of the Finnish population lives in poverty by the same standard. But 0.2% of Turkmenistan lives below its national poverty line (according to the CIA world factbook), and 13.7% of the Finland lives below its national poverty line.
Poverty Index is another statistic to be aware of. The wikipedia article on absolute poverty rate suggests that Poverty Index addresses some of the shortfalls of the Poverty Rate statistic.
Using a poverty threshold is problematic because having an income slightly above or below is not substantially different; the negative effects of poverty tend to be continuous rather than discrete, and the same low income affects different people in different ways. To overcome this problem, a poverty index or indices can be used instead
Its calculation is too complex to summarize here, but it takes into account survival rate through age 60, literacy skills, long-term unemployment, and population living below 50% of median income.
A caution against over-reliance on growth as an indicator.
Although economic growth is good, growth can lead to inflation (sometimes good, though some people consider it always bad), and inflation levels above a few percent can lead to hyperinflation (definitely bad).
Most economists today agree that 2.5-3.5% GDP growth per year is the most that our economy can safely maintain without causing negative side effects.
Over time, the growth in GDP causes inflation, and inflation begets hyperinflation. Once this process is in place, it can quickly become a self-reinforcing feedback loop. This is because in a world where inflation is increasing, people will spend more money because they know that it will be less valuable in the future. This causes further increases in GDP in the short term, bringing about further price increases. Also, the effects of inflation are not linear; 10% inflation is much more than twice as harmful as 5% inflation.
most economists today (including those in charge of U.S. monetary policy) agree that a small amount of inflation, about 1-2% a year, is more beneficial than detrimental to the economy.