It depends on the country in which it's written and also on the context.
While different style guides say different things, and individuals can also do things differently, I will answer based on common styling in both the US and the UK.
Given clarification in a comment to the question, this is about dialogue that's happened at different times. As such, I would consider what's being quoted is really a series of dialogue fragments rather than actual sentences. Therefore, I would treat these fragments as words as words (or phrases as phrases), which do not need dialogue tags.
I also want to slightly rephrase the sentence in the question to make the context a bit more clear.
In the US:
She has told me at different times "take me out," "let's travel together," and "I love ice cream."
In the UK:
She has told me at different times 'take me out', 'let's travel together' and 'I love ice cream'.
(Note that, unlike US English, UK English commonly doesn't use the serial comma—so I removed it.)
Of course, none of this is written in stone, and many people, in each country, will do things differently.
There are several variations to this that could be used. For instance, with words as words, quotation marks are sometimes eschewed altogether in favour of italics:
She has told me at different times take me out, let's travel together, and I love ice cream.
Here, it doesn't matter if it's US styling or UK styling. With italics, both would handle it the same way.
Also, in the US, an exception can be made to use UK styling (with single quotation marks and punctuation) if it's a scientific text or involves literary criticism. As per The Chicago Manual of Style (17th ed.), 6.9:
In an alternative system, sometimes called British style (as described in the New Oxford Style Manual), single quotation marks are used, and only those punctuation points that appeared in the original material are included within the quotation marks; all others follow the closing quotation marks. (Exceptions to the rule are widespread: for example, periods are routinely placed inside any quotation that begins with a capital letter and forms a grammatically complete sentence.) Double quotation marks are reserved for quotations within quotations. This system or a variation (like the one prescribed by Scientific Style and Format) may be appropriate in works of textual criticism or in computer coding and other technical or scientific settings.