The first input to a callback function is a handle to the callback object, which is often given the name hObject. On the other hand handles is the name that is often given to the struct that is saved with guidata. You can load or save that struct with the following lines.
%hObject is either a handle to the GUI figure, or an object that is a child of the GUI
My small guide to avoid GUIDE:
Make a figure (with f=figure;) and look into the doc for figure which properties you want to turn off (you probably want to set Menu and Toolbar to 'none')
Create buttons and axes and everything you need with functions like uicontrol and axes. Save the handles to each element to fields of a struct (like handles.mybutton=uicontrol(___);)
When you've finished loading all data (and saving it to fields of your handles struct), and creating all the buttons, save your handles struct to the guidata of your figure like this guidata(handles.f,handles);. (You can also use getappdata and setappdata)
You can set the Callback property of many objects. If you do, use a function name with an @ in front, or a char array that can be evaluated to valid code. (like @MyFunction or 'disp(''you pushed the button'')')
Callback functions will be called with two arguments: the first is a handle to the callback object, the second is eventdata that may contain special information. To get access to your data, just use handles=guidata(gcbo);. You can replace the gcbo function with the name of the first input to your callback function if you prefer.
More information about callbacks can be found in multiple places in the doc, for example here.