I am learning two foreign languages. I know that learning does not happen in a vacuum and that I need to practice with native speakers. Most natives in my area already speak English, and I have nothing else to offer. I am uncomfortable with the idea of befriending someone based on their race, in addition to the fact that I do not want to make them feel like I am using them.
You have a few options here.
Number one - If there are people you know already in a friendship or other connected way who speak the language you want to learn, be upfront with what you're doing. Mention that you want to practice your target language, and ask them if they're okay with that. If you have specific topics you want to talk about, mention that as well. This way, everyone is on the same page and they can decline or not as they choose. Who knows, you might already have friends who want to help!
Number two - if you know people in your area that might speak the language, but you're worried about it looking like you're befriending them based on race or other principles - again, just be honest. Mention that you know they know your target language (or ask if they know it, if you're not sure), and that you've been looking to practice your target language, and you wondered if they'd be okay with you speaking to them in whatever language you want to learn, and correcting you along the way. They might say yes, they might say no - but at least this way you all know what the expectations are of your interactions. This way it lessens the potential for someone to think you are just using them for free tutoring, because you've been honest and upfront about what you need and want. Be wiling to meet them halfway - maybe not all of your interactions will be language learning ones, but I'm sure you could gain other benefits from new friendships as well.
Number three - avoid this entirely by finding another resource - maybe there is a group in your area that already meets to do this very thing (like in my area, we have a number of "conversation circles" for English language learners, where they get together over coffee or the like knowing they're all there to practice English). Or look online - there are a number of resources for learning languages, and I know I've seen some sites out there specifically for speaking with people in a particular target language to learn it.
Number four - offer to pay the people you know to practice with you. Maybe it's not monetary, maybe you make them dinner once in a while and you share a meal and a conversation in a particular language, or you trade skills - I trade baking for rides a lot because I don't have a car. Perhaps you could do the same but for the language help.
For both number one and number two, tailor your expectations to what sort of help you're looking for. Do you want them to speak also in your target language back to you, or are they okay to use whatever language you already share? being as upfront and honest about what your goals are leaves the people you are looking to speak with free to make an informed decision about whether or not they want to take this on.
Just tell them you want to learn their language and you like to practice with people who are natives of that language.
You could just go to anyone of them and ask something like: Hi, I like to learn Spanish and I like to practice. Would you like to talk to me or do you know someone who I should talk to?
Many people are happy that others are interested in them and in their language - without any compensation. Just try!
As a person who has studied several foreign languages, I completely understand your desire to work with native speakers.
A large part of the answer to the question centers on location. If you encounter native speakers in the United States, they may want to speak mainly English with you. This is not to dismiss their culture; it is to help them assimilate as a lot of immigrants believe that learning English is critical to becoming functioning members of American society.
If you're in their country, then just go ahead and speak it! Most of the speakers there understand you're trying to learn and, in my experience, will happily correct you if you ask them. I've heard that several times: "I understand what you are trying to say, but we say it like this:..."
If you befriend someone solely on their ability to help you, that would come across as manipulative or insincere. If there are groups that get together, perhaps you could join them? They may want to speak their native tongue together as a bonding experience and may welcome someone who wants to learn to communicate with them.
Being open and clear to people solves both your problems.
If you're interested in learning a new language, you've got to be able to show your enthusiasm to the native speakers, and tell them exactly what you're doing. Treat them as you'd treat any other friends.
Most people, if they're proud of their native language and culture, will be more than happy to help another person trying to learn their language.
If you feel guilty about it, or that you owe them something, offer them either money, or payment in kind, like inviting along for movies, giving them little things they'll surely like, etc. It's not like teaching them another language is the only way of returning their favor...
However, not everyone may be completely open to helping out another person, and some may feel offended, as you have noted, for being befriended for their race. In this case, tell them you meant well, but stay clear of them. They're not the kind who'll be helpful for your needs.