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I have tried some of the solutions to similar questions, but none of them included minipages, which seems to be the problem (I`m fairly new to LaTeX)

So, here's the coding:

% Definition of circles
deffirstcircle(0,0) circle (1.5cm)
defsecondcircle(0:2cm) circle (1.5cm)

colorletcircle edgeblack!50
colorletcircle areagrey!20

tikzsetfilled/.style=fill=circle area, draw=circle edge, thick,
 outline/.style=draw=circle edge, thick

setlengthparskip5mm

% Set A or B
begintikzpicture
beginminipage[t][2cm][t].48textwidth
 draw[filled] firstcircle node $A$
 secondcircle node $B$;
 node[anchor=south] at (current bounding box.north) $M$;
endminipage
endtikzpicture
% Set A and B
begintikzpicture
beginminipage[t][2cm][t].48textwidth
 beginscope
 clip firstcircle;
 fill[filled] secondcircle;
 endscope
 draw[outline] firstcircle node $A$;
 draw[outline] secondcircle node $B$;
 node[anchor=south] at (current bounding box.north) $M$;
 node[anchor=west] at (current bounding box.east) $A cap B$
endminipage
endtikzpicture

Also, if you know how to put the "B" in the center of the circle on the right I`d appreciate your help! Thank you.

Welcome to TeX.SE! It is good that you included your code. In terms of drawing a rectangle around the venn diagram, you could import the fit library from Tikz using usetikzlibraryfit. I have completed an example to illustrate. Also, I have changed the code a little bit. You now have one tikzpicture (with no minipage). The first venn diagram has its own scope and the second has its own as well. I have changed the placement of M so that it would be easier to read. There are two types overall in terms of appearance, you choose the one you like.

documentclass[margin=1cm, tikz]standalone
usepackagetikz,xcolor,color
usetikzlibraryfit
begindocument

tikzsetfilled/.style=fill=circle area, draw=circle edge, thick,
 outline/.style=draw=circle edge, thick

setlengthparskip5mm

begintikzpicture
 % Definition of circles
 deffirstcircle(0,0) circle (1.5cm)
 defsecondcircle(0:2cm) circle (1.5cm)
 %
 colorletcircle edgeblack!50
 colorletcircle areagray!20
 %
 beginscope[local bounding box = orScope]
 draw[filled] firstcircle node $A$
 secondcircle node $B$;
 node[anchor=south] at (orScope.north)$M$;
 node[anchor=north] at (orScope.south) $A cup B$;
 endscope
 node[fit=(orScope), draw] ;% The frame around the scope
 %
 beginscope[xshift = 7cm,local bounding box = andScope]
 beginscope
 clip firstcircle;
 fill[filled] secondcircle;
 endscope
 draw[outline] firstcircle node $A$;
 draw[outline] secondcircle node $B$;
 node[anchor=south] at (andScope.north) $M$;
 node[anchor=west] at (andScope.south east) $A cap B$;
 endscope
 node[fit=(andScope), draw] ;
endtikzpicture

enddocument

Another solution:

If you really wants to use minipage, you have to put the environment outside tikzpicture. Literally minipage does nothing for you here, but it does many things against you.

However, for figures side-by-side, a subfigure environment (from subcaption) is better in this case.

And, to draw a rectangle, you can use the special node current bounding box, with some shifts if needed.

Btw your diagrams are too wide for the normal margin.

documentclassarticle
usepackage[margin=1in]geometry
usepackagetikz
% Definition of circles
deffirstcircle(0,0) circle (1.5cm)
defsecondcircle(0:2cm) circle (1.5cm)

colorletcircle edgeblack!50
colorletcircle areagray!20

tikzsetfilled/.style=fill=circle area, draw=circle edge, thick,
 outline/.style=draw=circle edge, thick

setlengthparskip5mm
begindocument
noindentbeginminipage[t].5textwidth
centering%
begintikzpicture
 draw[filled] firstcircle node $A$
 secondcircle node $B$;
 node[anchor=south] at (current bounding box.north) $M$;
endtikzpicture
endminipage%
beginminipage[t].5textwidth
centering%
begintikzpicture
 beginscope
 clip firstcircle;
 fill[filled] secondcircle;
 endscope
 draw[outline] firstcircle node $A$;
 draw[outline] secondcircle node $B$;
 node[anchor=south] at (current bounding box.north) $M$;
 node[anchor=west] at (current bounding box.east) $A cap B$;
 draw ([shift=(-1ex,1ex)]current bounding box.north west) rectangle
 ([shift=(1ex,-1ex)]current bounding box.south east);
endtikzpicture
endminipage
enddocument

There are dedicated packages for that. Assuming you do not want to use any of those, I'd like to advertize some styles that help you drawing these diagrams. These are not inside, inside to specify the sets and frame for drawing a frame around these pictures. Your first picture is easy, you only need to fill the circles. The second one can be achieved with

pgfkeysinside/.list=pathA,pathB,shade=gray

and the frames can be done with

draw[frame=5pt];

where 5pt indicates the margin.

Here are code and result.

documentclassarticle
usepackagetikz
usetikzlibrarybackgrounds
% based on https://tex.stackexchange.com/a/12033/121799
tikzsetreverseclip/.style=insert path=(current bounding box.south west)rectangle 
(current bounding box.north east) , 
use path/.code=pgfsetpath#1,%learned from Kpym
frame/.style=insert path=
([xshift=-#1,yshift=-#1]current bounding box.south west) rectangle
([xshift=#1,yshift=#1]current bounding box.north east)

begindocument
pgfkeysnot inside/.code=clip[use path=#1,reverseclip];,
inside/.code=clip[use path=#1];,
shade/.code=fill[#1] (current bounding box.south west)rectangle 
(current bounding box.north east);
begintikzpicture
draw[thick,fill=gray] (-1,0) node$A$ circle [radius=1.5cm]
(1,0) node$B$ circle[radius=1.5cm] (0,1.5) node[above]$M$
(0,-1.5) node[below]$Acup B$;
draw[frame=5pt];
endtikzpicture~%
begintikzpicture
draw[thick,save path=pathA] (-1,0) node$A$ circle [radius=1.5cm];
draw[thick,save path=pathB] (1,0) node$B$ circle[radius=1.5cm];
path (0,1.5) node[above]$M$
(0,-1.5) node[below]$Acap B$;
beginscope[on background layer]
pgfkeysinside/.list=pathA,pathB,shade=gray
endscope
draw[frame=5pt];
endtikzpicture
enddocument