By now several Japanese people have told me that they also find it difficult and annoying to read texts without kanji.
On top of what the others said (learn tons of words and get used to grammar), I can think of several options for reading:
If you look at books aimed at first year schoolchildren, they come with gaps between the words to help them along. But they have basically no kanji, so that might not be ideal either.
The Absolute Beginner Book Club will soon (EDIT: actually about now) start reading a book aimed at second year schoolchildren, that might be an option as well.
There are also Graded Readers, which are books geared towards foreign language learners, and they also come with gaps between the words in the lower levels. They also use simple grammar and vocab (depending on the level) so that might be your best option for this early stage.
Another option would be to jump in at the deep end and to get a book that is geared towards middle schoolers and upwards; there are editions that come with full furigana so you need not worry about the kanji you don’t know yet. But you would need a solid foundation in grammar (at least Genki I + II, more would be even better) and of course you will need to look up words galore, so it will be a very very bumpy ride. If you go to BookWalker or the like, you can always look at the first pages of a book, so you can try it out and see how far you get before investing any money.
You can also start with reading manga; they are less text-heavy but can still offer enough text to read. There are tons of suggestions in the list of book clubs, and you can look at the old threads to find answers to your questions (and even still ask new questions).
If you need more inspiration, please make sure to take a look at Resources for Starting to Read Japanese Content !