With callbacks, you always "respond" via one function. For example:
function getUsers (age, done) {
    // done has two parameters: err and result
    return User.find({age}, done)
}
Promises lets you respond according to the current state:
function getUsers (age) {
    return new Promise((resolve, reject) => {
        User.find({ age }, function (err, users) {
            return err ? reject(err) : resolve(users) 
        })
    })
}
This flattens the "callback pyramid." Instead of
getUsers(18, function (err, users) {
    if (err) {
        // handle error
    } else {
        // users available
    }
})
You can use:
getUsers(18).then((users) => {
    // `getPetsFromUserIds` returns a promise
    return getPetsFromUserIds(users.map(user => user._id))
}).then((pets) => {
    // pets here
}).catch((err) => {
    console.log(err) // handle error
})
So, to answer your question, first you'd want to use a promise for your http requests:
function GET (url) {
    return new Promise((resolve, reject) => {
        let xhr = new XMLHttpRequest()
        xhr.open('GET', url, true)
        xhr.onload = function () {
            if (this.status >= 200 && this.status < 300) {
                return resolve(xhr.response)
            } else {
                return reject({ status: this.status, text: xhr.statusText })
            }
        }
        xhr.onerror = reject
        xhr.send()
    })
}
Then, you'd want to incorporate this into your loadRates function:
function loadRates (days) {
    var URL = URL_GENERATOR(days)
    return GET(URL).catch((err) => {
        // handle our error first
        console.log(err)
        // decide how you want to handle a lack of data
        return null
    }).then((res) => {
        localStorage.setItem('rates' + days, res)
        return res
    })
}
Then, in initSession:
function initSession () {
    Promise.all([ loadRates(0), loadRates(10) ]).then((results) => {
        // perhaps you don't want to store in local storage,
        // since you'll have access to the results right here
        let [ zero, ten ] = results
        return buildTable()
    })
}