Neither of OP's examples are very "natural".
The first is particularly unlikely because native speakers don't normally give a try to do anything. If some course of action has already been mentioned or is implied by the circumstances, we say...
1: "I'll give it a try!"
2: "I'll give it a go!"
3: "I'll give it a shot!"
It might help you remember the need for that "it" if you ask yourself what exactly the construction would mean without it. If you are giving the try/go/shot, who or what receives that "effort/attempt"? Logically, the "recipient" must be either the action you intend doing, the problem that action is intended to solve, or the solution itself. It doesn't really make any difference which you imagine "it" to be - the overall meaning is the same - but it does need to be there.
The second is also rather unlikely, because "I will try" normally means the same as "I will do my best". Sometimes we do repeat things in different ways to add emphasis, but in a case like this it would be more natural to say...
4: "I [really] will try [very] hard!"
5: "I will do my [very] best."
Note that 1-3 are all relatively informal (go and shot more so than try). And in most constructions like this, including really for added emphasis is also rather informal. But take particular note of #5, because that's almost certainly your best bet in formal or semi-formal contexts (an email to a customer assuring him that you will make every effort to resolve his problem, for example). But provided you contract "I will" to "I'll", it wouldn't sound starchy or stilted in informal contexts, so it's a good one to know.
As regards OP's supplementary question (what other verbs can be combined with "give"), the answer is that no "verbs" can be used in this way - only nouns (a try, a shot, a go, etc.).
The already-mentioned "targeted effort" nouns are the things most commonly given, but the noun forms of various "verbs of perception" can also be used it this construction. For example...
6: "This new album is great! You should give it a listen!"
7: "I hear 'House of Cards' is supposed to be good. Maybe I'll give it a look!"
8: "Sushi is actually quite nice. Here [proffering a mouthful] - give it a taste"
Because the format is actually quite "productive", you will actually come across things like...
9: "You should give it a watch."
...but you need to be a bit careful there. Many native speakers would say that's at least "iffy" - although it seems to fit the general pattern of #6-8, a watch isn't really a valid noun derived from to watch like that. Strictly speaking the same applies to a listen (which you can normally only give or have, in the type of context we're looking at here). A listen is okay because it doesn't clash with any existing noun sense. But a watch is problematic because it's so commonly associated with the wrist-worn timepiece.