You are wrong if you believe the caricature of conservative that is portrayed regularly on this site. Not that they don't exist but they are a minority. Regular conservatives don't hate minorities, the poor, gays etc. Most just want to go about their daily lives. Regular conservatives don't post on conservatives sites and aren't activists.
I have changed my views on a few things thanks to my smart liberal friends so I'm going to write about those and a few other issues so that you can try to find a middle ground with your conservative friends and family.
If you believe that all conservatives are racists (or any more than liberals are) hate americans, don't believe in the constitution and eat babies then don't read below the fold. But if you believe that some conservatives are reasonable, but just have a different perspective then keep reading.
Heck I'm an uber conservative and I voted for Obama twice and might consider voting for Bernie Sanders.
The case of the Colorado bakery refusing service for a gay wedding lit conservative and liberal forums alike on fire. I'm sure you are familiar with it so I won't rehash it here.
Conservatives everywhere claimed that the baker should have the freedom to associate and do business with whoever they want. That it is a natural right to do so. They felt it was an infringement on people's freedom of religion.
Here is what they are missing that you can explain to your conservative friends and family.
A business is a special entity. It gets many legal benefits from our society and as a result also has some restrictions.
Business benefits:
1. No taxes on expenses. When you and I make 100K we pay $25K (or so) on that 100K even if we spend every penny to live our lives. When a business makes 100K the business only pays taxes on the leftover after expenses. We take it for granted that it should be so, but there is no logical reason that a business shouldn't pay tax on revenue. It just happens to be a huge benefit that the fictional entity of a business receives.
2. Businesses are standalone entities. This fiction is supported by our legal framework. They can have shareholders and have an entire legal system around supporting their existence and of having shareholders and protecting those shareholders to increase investment.
a. Shareholder's ownership is protected by law and therefore they are more likely to invest.
b. Shareholders have no liability due to malfeasance on the part of the company
c. Shareholders are protected from the debts of the company including during bankruptcy
3. Businesses can enter into contracts that shareholder's are not on the hook for. The legal system ensures that when a fictional entity like a business enters into a contract, the business has recourse if the contract is violated. If a customer does not pay, the business can go after the customer using the courts. Businesses don't actually exist except in the legal system.
There are many other benefits given to the fictional entity of a business. In return there are many restrictions, one of which is who they must service.
One of the key gaps is when a baker is an owner AND an employee it can be confusing as to who is being forced to perform the service. The only entity that must provide the service is the business.
A business (fictional entity) must provide the service. It doesn't matter which employee within the business provides that service. An employee can refuse to perform the service (at the risk of getting fired), but a business entity simply cannot. Things become confusing when the only employee is also the owner, yet the business still must provide the service. The owner/employee could conceivably subcontract out the service to someone else so they don't personally have to perform the service, but the business must follow the law and provide services to protected classes. The employee does not have to perform the service, but the business does.
If a person chooses to provide services without receiving all the benefits of a business, then many of the restrictions are removed. They then are a private club or have a hobby. There is no business entity created and they don't receive many of the benefits above. For example if someone decides not to pay them, they (should) have no recourse. Lease agreements should be done as an individual and there should be no bankruptcy protection from the debts of the non-business.
I have tried this logic with several recalcitrant conservatives and a few simply won't agree, they simply repeat "freedom of association" over and over. But a larger number that I have debated with have been persuaded.
Like many libertarians some people want all the benefits of protection of the law but don't want to provide protection of the law to others. Usually this is because they are blind to the fact that the protections even exist.
Good luck!