Hi Soffy,

For confusing things like this, I really recommend just following the example instead of analysing it too much. The problem is, both <em>to</em> and <em>for</em> have about twenty meanings each, depending on context. Look too closely and you'll just become lost.

Anyway, I'll try to explain a little bit.

<em>My English goal for July 9th, 2015</em> - this means your plan on that day, or what you plan to do when that day comes.  <em>To</em> will change the meaning and it will (awkwardly) sound like "until".  Remember that <em>to</em> can mean a transit somewhere or somehow - movement or change.

We use <em>is</em> because we understand the message as, "My English goal is..."

Verbs in the infinitive form (<em>to increase</em>) can mean an intention or an action you wish for. An expanded vocabulary is what you want, so <em>to increase</em> tells us your intention.  "For increase" is simply wrong. If you have a preposition (apart from <em>to</em> - more rules for that!) the action that follows must be in a gerund form, so <em>for increasing</em> is technically correct. However, now the sentence changes meaning again (and sounds awkward again), because we use <em>for (verb)ing</em> to mean an existing purpose.

So as I said, it's far easier to just accept the example as correct and follow that pattern.