Step 1 - Definitions of Many Systems
My approach tends to require an audience with a long attention span. But I like to start by explaining Liberal Democracy for background information, because it will be used later. Then explain Authoritarian Socialism, and how all of the Socialist countries that Americans / conservatives look down on have been Authoritarian Socialist, e.g. Soviet Russia, Maoist China, Cuba, and Venezuela (in contemporary cases, you can even prove who is / isn't authoritarian using the Democracy Index). Then explain what Democratic Socialism and Social Democracy are, making note of the economic distinctions between them (i.e. Social Democracy is a free market system with some regulations, taxation, and social spending; whereas Democratic Socialism is a truer form of Socialism with broader public ownership of the means of production). Point out that both Social Democracy and Democratic Socialism respect the beliefs of Liberal Democracy (exception being that Democratic Socialism disagrees with one of the many tenets -- the free market economic aspect). Point out that Authoritarian Socialism does not respect the tenets of Liberal Democracy.
Step 2 - Who is Bernie Sanders
Then point out that Bernie Sanders has mislabeled himself as Democratic Socialist, but is really Social Democratic, as evidenced by the fact that he has not pushed for broad public ownership of many industries, intends to keep the free market in place, but intends to raise taxes on it and expand social spending.
Step 3 - Social Democratic Success Stories (United States)
At this point, you will need to make a case that Social Democracy has some sort of successful track record. I would start by pointing to the 1930's - 1970's United States, in which the top marginal tax rates were 70-94% instead of 37% (you may be forced to explain how marginal tax rates work, the estate tax was nearly 80% instead of 40% (you may be forced to explain that only estate amounts above a certain qualified amount are taxed), the statutory corporate tax rate was about 30% higher, and the effective corporate tax rate was about 20% higher. Correlate those tax rates with positive / inverse economic indicators of prosperity like the public debt (as a % of GDP), unemployment rate, poverty rate, federal budget balance, wealth inequality, and inflation-adjusted minimum wage to show that our time of greatest economic success was around 1969, and that our economy has consistently gotten worse as we've slashed tax rates and implemented Reaganomic policies.
Step 4 - Social Democratic Success Stories (Abroad)
Various forms of Social Democracy (Social Capitalism, Social Democracy, Democratic Socialism, and Mixed Economies) have produced success stories abroad. Some examples of these economies that have greater general welfare and prosperity than America include Switzerland, Norway, Denmark, Sweden, Australia, Iceland, New Zealand, Germany, Netherlands, Czech Republic, Austria, United Kingdom, Finland, Canada, and Ireland. I'm gauging country rankings based on Education, GDP Per Capita (America's best stat), Murder Rate, Public Debt Per Capita, Income Inequality (Gini Index), Unemployment Rate, Social Progress Index, World Happiness Report, Index of Economic Freedom, and Democracy Index.
Step 5 - Explain Economic Velocity
Also known as velocity of money, and nearly synonymous with consumer spending; economic velocity dictates whether we are in a recession or a growth period. It helps dictate the unemployment rate, the number of people working part-time vs full-time, the poverty rate, quality of life, and the number of people drawing on government assistance. The more that wealth pools in the hands of the wealthy, the more that economic velocity slows down and recession sets-in (this is because the upper class does not spend money they have at as high a rate as the lower class does). The more that wealthy citizens and corporations are taxed, returning that wealth to the lower and middle classes, the more that they are going to spend the money, translating to greater economic velocity, creating more private sector jobs, increasing work week hours for part-time workers, increasing tips for people who work for tips, and delivering all of the other benefits that correlate with a growth phase. Returning wealth to the lower and middle class creates more opportunities for the middle class to engage in capital investment and start small businesses; so that not all new wealth created is going strictly to the wealthiest citizens. Proponents of the Laffer Curve claim that increasing taxes will cause economic slowdown, but they miss that the rules of the Laffer Curve only apply under a flat tax, not under a tiered tax system with marginal rates, which is what the United States has. When wealth is taxed from the classes who spend their money the least, and given to classes that spend their money most readily, the effect is the exact opposite of what the Laffer Curve predicts, i.e. this will cause economic acceleration.
Step 6 - Appeal to Impetus of Automation
The promise of automation was that it could one day reduce the amount that we have to work and create a post-scarcity society. But as long as corporations are possessing the profits of automation, it means that jobs are continually getting eliminated while poverty and wealth inequality gets continually worse. If we taxed wealthy corporations more, implemented a guaranteed income that was proportional to the scope of automation (and/or what the federal budget could afford), and reduced the hours in the work week to be proportional to the amount of work that needed doing; then we should always be able to make a living wage, maintain low unemployment, and work fewer and fewer hours over time while stemming the tide of growing wealth inequality. Conservatives tend to believe that jobs eliminated in America have mostly been outsourced or taken by immigrants, so now would be a good time to point out that 87% of jobs lost in America were eliminated by automation, compared to only 13% that were eliminated due to outsourcing.