I third what Mike wrote: "one definitely needs a solid grounding in 1-category theory before learning higher category theory". With that being said, elaborating and expounding upon janed0e's suggestion, what follows are two study plans according to the prior knowledge of the student. Of course, there is no canonical way to approach learning higher category theory, so adjust the readings as needed. Note well, following the modern terminology as developed by Joyal, quasicategories are a model for ($\infty$, 1)-categories. Following the modern terminology as developed by Lurie, the unqualified usage of '$\infty$-category' or '$\infty$-categories' designates '($\infty$, 1)-category' and '($\infty$, 1)-categories', respectively.
Assumption: Student has no knowledge of 1-category theory (or simplicial sets) and wishes to get the flavor of infinity-category theory, without getting bogged down by technical details, in as short a time as can be reasonably expected. The implicit assumption is that the student has a budget of zero dollars.
Possible reading material and sequence with which to read:
0) J. Adamek, H. Herrlich, G. Strecker: Abstract and Concrete Categories: The Joy of Cats
1) G. Friedman: An elementary illustrated introduction to simplicial sets
2) J. Lurie: What is ... an $\infty$-Category?
3) M. Boyarchenko: Notes and Exercises on $\infty$-categories
4) M. Groth: A Short Course on $\infty$-categories(http://www.math.ru.nl/~mgroth/preprints/groth_scinfinity.pdf)
Repeating what Giorgio Mossa wrote, (0) has an abundant number of examples from topology, algebra, and theoretical computer science. As Mike Shulman noted, (0) is rather idiosyncratic. (0) uses the term 'quasicategory' for what Mac Lane called metacategories. See the nLab page metacategory (http://ncatlab.org/nlab/show/metacategory) for further clarification about the terminology clash. (0) can be supplemented with video lectures by the Catsters (http://www.scss.tcd.ie/Edsko.de.Vries/ct/catsters/linear.php) and Wikipedia's Outline of category theory (http://en.wikipedia.org/wiki/Outline_of_category_theory).
Assumption: Student has knowledge of 1-category theory (but not simplicial sets) and wishes to get an in depth experience of infinity-category theory, allowing an 'ample' amount of time.
Possible reading material and sequence with which to read:
0) P. G. Goerss and J. F. Jardine: Simplicial Homotopy Theory (http://dodo.pdmi.ras.ru/~topology/books/goerss-jardine.pdf)
1) J. Lurie: What is ... an $\infty$-Category?
(http://www.ams.org/notices/200808/tx080800949p.pdf)
2) M. Boyarchenko: Notes and Exercises on $\infty$-categories (http://www.math.uchicago.edu/~mitya/langlands/quasicategories.pdf)
3) M. Groth: A Short Course on $\infty$-categories
(http://www.math.uni-bonn.de/~mgroth/InfinityCategories.pdf)
4) J. Lurie: On the Classification of Topological Field Theories
(http://arxiv.org/abs/0905.0465)
5) C. Simpson: Homotopy Theory of Higher Categories
(http://hal.archives-ouvertes.fr/docs/00/44/98/26/PDF/main.pdf)
6) J. Lurie: Higher Topos Theory
(http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf)
(4) may be a more readable than (6), since (4) is an expository paper that gives an
informal account of the classification of topological field theories using the technology of
($\infty$, n)-categories. (4) can be nicely supplemented by Lurie's video lecture series
on "Topological Quantum Field Theories and the Cobordism Hypothesis" (http://lab54.ma.utexas.edu:8080/video/lurie.html), as well as the corresponding
notes for said lecture (http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/perspectives_TQFT_notes.html).
(5) offers a broad perspective of current research in higher category theory.
(6) develops in detail the vast generalization of 1-category theory to ($\infty$, 1)-category theory.
For further roadmaps on learning higher category theory, look at this nForum discussion on reading Lurie's Higher Topos Theory (http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2748&page=1#Item_0).
Hope this helps.
