Vrftjkry

That is what I have so far:

documentclass[tikz,border=3.14mm]{standalone}

begin{document}

usetikzlibrary{intersections,decorations}



begin{tikzpicture}[thick,

    dot/.style = {

        draw,

        fill = black,

        circle,

        inner sep = 0pt,

        minimum size = 2pt

}]



    draw[smooth cycle, tension=0.6] plot coordinates{(-1,-.4) (1.1,-.6) (.9,.5) (-1.1,.5)} node at (1,.6) {$Omega$};

    draw[blue] (-1.1,.5) .. controls (-.5,-.2) and (.5,0) .. coordinate[dot,pos=0.3,black] (A) coordinate[dot,pos=0.5,black] (B) coordinate[dot,pos=0.7,black] (C) coordinate[pos=0.92,black] (D) (1.1,-.6);

    draw[gray] (A) circle (.3);

    draw[gray] (B) circle (.375) node[black,anchor=north] {$x_0$};

    node[anchor=south,gray] at ([yshift=10]B) {$gamma(bar{D})$};

    draw[gray] (C) circle (.3);

    node[blue,anchor=south west] at (D) {$S$};



end{tikzpicture}



end{document}

which looks like

Now, I want to add two dashed lines at each side of S going from the beginning and end of S touching the circles exactly such that each circle stays within.

Something like:

UPDATE: Using tangents, see e.g. this nice answer.

documentclass[tikz,border=3.14mm]{standalone}

usetikzlibrary{calc}

begin{document}



begin{tikzpicture}[thick,

    dot/.style = {

        draw,

        fill = black,

        circle,

        inner sep = 0pt,

        minimum size = 2pt

}]



    draw[smooth cycle, tension=0.6] plot coordinates{(-1,-.4) (1.1,-.6) (.9,.5) (-1.1,.5)} node at (1,.6) {$Omega$};

    draw[blue] (-1.1,.5) coordinate(start) .. controls (-.5,-.2) and (.5,0) ..

    coordinate[dot,pos=0.3,black] (A) coordinate[dot,pos=0.5,black] (B)

    node[black,anchor=north] {$x_0$} coordinate[dot,pos=0.7,black] (C) coordinate[pos=0.92,black] (D) (1.1,-.6)

    coordinate(end);

    % see https://tex.stackexchange.com/a/76226/121799

    pgfmathsetmacro{rsmall}{0.6}

    pgfmathsetmacro{rbig}{0.75}

    node [draw=gray,circle,minimum size=rsmall*1cm] (cA) at (A){};

    node [draw=gray,circle,minimum size=rbig*1cm] (cB) at (B){};

    node[anchor=south,gray] at ([yshift=10]B) {$gamma(bar{D})$};

    node [draw=gray,circle,minimum size=rsmall*1cm] (cC) at (C){};

    node[blue,anchor=south west] at (D) {$S$};

    begin{scope}[overlay] % see https://tex.stackexchange.com/a/76226/121799

    coordinate (cAB) at (barycentric cs:B=-rsmall,A=rbig);

    coordinate (cCB) at (barycentric cs:B=-rsmall,C=rbig);

    end{scope}

    foreach Y in {1,2}

    {foreach X in {A,C}

    {path (tangent cs:node=cX,point={(cX B)},solution=Y) coordinate(pBX-Y)

    (tangent cs:node=cB,point={(cX B)},solution=Y) coordinate(pX B-Y);}

    path (tangent cs:node=cA,point={(start)},solution=Y) coordinate(pstartA-Y)

    (tangent cs:node=cC,point={(end)},solution=Y) coordinate(pendC-Y); }  

    draw[red,dashed] plot[smooth] coordinates {(start) (pstartA-2) (pBA-2) (pAB-2) (pCB-1)  (pBC-1)

    (pendC-1) (end)};

    draw[red,dashed] plot[smooth] coordinates {(start) (pstartA-1) (pBA-1) (pAB-1) (pCB-2)  (pBC-2)

    (pendC-2) (end)};

end{tikzpicture}

end{document}

OLDER ANSWER: You started drawing this with Bezier curves (which I may not necessarily have done), so perhaps the best way is to continue this strategy. Yet I think it helps to make the circles nodes such that you can easily access the points on their peripheries.

documentclass[tikz,border=3.14mm]{standalone}

begin{document}



begin{tikzpicture}[thick,

    dot/.style = {

        draw,

        fill = black,

        circle,

        inner sep = 0pt,

        minimum size = 2pt

}]



    draw[smooth cycle, tension=0.6] plot coordinates{(-1,-.4) (1.1,-.6) (.9,.5) (-1.1,.5)} node at (1,.6) {$Omega$};

    draw[blue] (-1.1,.5) .. controls (-.5,-.2) and (.5,0) ..

    coordinate[dot,pos=0.3,black] (A) coordinate[dot,pos=0.5,black] (B)

    node[black,anchor=north] {$x_0$} coordinate[dot,pos=0.7,black] (C) coordinate[pos=0.92,black] (D) (1.1,-.6);

    node [draw=gray,circle,minimum size=0.6cm] (cA) at (A){};

    node [draw=gray,circle,minimum size=0.75cm] (cB) at (B){};

    node[anchor=south,gray] at ([yshift=10]B) {$gamma(bar{D})$};

    node [draw=gray,circle,minimum size=0.6cm] (cC) at (C){};

    node[blue,anchor=south west] at (D) {$S$};

    draw[red,dashed] (-1.1,0.5) to[out=-20,in=190] (cA.110)

    to[out=10,in=170] (cB.90) to[out=-10,in=170] (cC.80) to[out=-10,in=150] (1.1,-.6); 

    draw[red,dashed] (-1.1,0.5) to[out=-50,in=150] (cA.250)

    to[out=-20,in=180] (cB.270) to[out=00,in=190] (cC.280) to[out=10,in=180] (1.1,-.6); 

end{tikzpicture}

end{document}

An empirical solution:

documentclass[tikz,border=3.14mm]{standalone}

begin{document}

begin{tikzpicture}[thin,

    dot/.style = {

        draw,

        fill = black,

        circle,

        inner sep = 0pt,

        minimum size = 2pt

}]



    draw[smooth cycle, tension=0.6] plot coordinates{(-1,-.4) (1.1,-.6) (.9,.5) (-1.1,.5)} node at (1,.6) {$Omega$};

    draw[blue] (-1.1,.5)coordinate(s1) .. controls (-.5,-.2) and (.5,0) .. coordinate[dot,pos=0.3,black] (A) coordinate[dot,pos=0.5,black] (B) node[black,below]{$x_0$} coordinate[dot,pos=0.7,black] (C) coordinate[pos=0.92,black] (D) (1.1,-.6)coordinate(s2);

    draw (A) node[draw,minimum size=0.52cm,circle] (C1){};

    draw (B) node[draw,minimum size=0.65cm,circle] (C2){};

    draw (C) node[draw,minimum size=0.52cm,circle] (C3){};

    node[blue,anchor=south west] at (D) {$S$};

    node[anchor=south,gray] at ([yshift=10]B) {$gamma(bar{D})$};

    draw[dashed,smooth, tension=0.5] plot coordinates{(s1) (C1.87)  (C2.85)  (C3.60) (s2)};

    draw[dashed,smooth, tension=0.5] plot coordinates{(s2) (C3.267) (C2.270) (C1.240) (s1)};    

end{tikzpicture}

end{document}