How do I bold (or format) a piece of text within a paragraph?

Question

You should use the RichText widget.

A RichText widget will take in a TextSpan widget that can also have a list of children TextSpans.

Each TextSpan widget can have a different TextStyle.

Here is the example code to render: Hello World

var text = new RichText(
  text: new TextSpan(
    // Note: Styles for TextSpans must be explicitly defined.
    // Child text spans will inherit styles from parent
    style: new TextStyle(
      fontSize: 14.0,
      color: Colors.black,
    ),
    children: <TextSpan>[
      new TextSpan(text: 'Hello'),
      new TextSpan(text: 'World', style: new TextStyle(fontWeight: FontWeight.bold)),
    ],
  ),
 );

[UPDATE]

The below answer fits best for couple of words and not for a paragraph,If you have a long sentence or a paragraph where you need to format a particular text prefer using RichText as suggested by @DvdWasibi in the above answer

[OLD ANSWER]

I like keeping my code short and clean this is How I Would do it add two text fields in a row one with Normal font and another bold,

Note: This may not look good for a long paragraph looks good for Headlines etc.

Row(children: <Widget>[
      Text("Hello"),
      Text("World", style: TextStyle(fontWeight: FontWeight.bold))
    ])
`

and you should get a desired output as "Hello World"

return RichText(
  text: TextSpan(
    text: 'Can you ',
    style: TextStyle(color: Colors.black),
    children: <TextSpan>[
      TextSpan(
        text: 'find the',
        style: TextStyle(
          color: Colors.green,
          decoration: TextDecoration.underline,
          decorationStyle: TextDecorationStyle.wavy,
        ),
        recognizer: _longPressRecognizer,
      ),
      TextSpan(text: 'secret?'),
    ],
  ),
);

Regex

You can use this widget. The example below always make numbers bold.

import 'package:flutter/material.dart';

class TextBold extends StatelessWidget{
  final String text;
  final String regex;
  static const _separator = " ";

  const TextBold({Key key, this.text, this.regex = r'\d+'}) : super(key: key);

  @override
  Widget build(BuildContext context) {

    final parts = splitJoin();

    return Text.rich(TextSpan(
        children: parts.map((e) => TextSpan(
            text: e.text,
            style: (e.isBold)
                ? const TextStyle(fontFamily: 'bold')
                : const TextStyle(fontFamily: 'light')))
            .toList()));
  }

  // Splits text using separator, tag ones to be bold using regex
  // and rejoin equal parts back when possible
  List<TextPart> splitJoin(){
    assert(text!=null);

    final tmp = <TextPart>[];

    final parts = text.split(_separator);

    // Bold it
    for (final p in parts){
      tmp.add(TextPart(p + _separator,p.contains(RegExp(regex))));
    }

    final result = <TextPart>[tmp[0]];
    // Fold it
    if (tmp.length>1) {
      int resultIdx = 0;
      for (int i = 1; i < tmp.length; i++)
        if (tmp[i - 1].isBold != tmp[i].isBold) {
          result.add(tmp[i]);
          resultIdx++;
        }
        else
          result[resultIdx].text = result[resultIdx].text
              + tmp[i].text;
    }

    return result;
  }
}

class TextPart{
  String text;
  bool isBold;

  TextPart(this.text, this.isBold);
}